Use the precise definition of a limit to prove the following limi...

Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

f(−1)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^− f(x)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^+ f(x)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1 f(x)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

f(1)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→1 f(x)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^− f(x)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^+ f(x)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3 f(x)

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Textbook Question

The following table gives the position $s\left(t\right)$ of an object moving along a line at time $t$ . Determine the average velocities over the time intervals $\left\lbrack1,1.01\right\rbrack$ , $\left\lbrack1,1.001\right\rbrack$ , and $\left\lbrack1,1.0001^{}\right.]$ . Then make a conjecture about the value of the instantaneous velocity at $t=1$ . <IMAGE>

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Textbook Question

Given the function $f\left(x\right)=-16x^2+64x$ , complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ , for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

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Textbook Question

Given the function $f\left(x\right)=-16x^2+64x$ , complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ as $x$ approaches $2$ .

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Textbook Question

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$ . Find the average velocity of the object over the following intervals.

$\left\lbrack1,4\right\rbrack$

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Textbook Question

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$ . Find the average velocity of the object over the following intervals.

$\left\lbrack1,2\right\rbrack$

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Textbook Question

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$ . Find the average velocity of the object over the following intervals.

$\left\lbrack0,3\right\rbrack$

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Textbook Question

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$ . Find the average velocity of the object over the following intervals.

$\left\lbrack0,h\right\rbrack$ , where $h\gt{0}$ is a real number

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

$\lim_{x \to 1^{-}} f (x)$

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

$\lim_{x \to 1^{+}} f (x)$

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

a. f(1)

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

d. $lim over x right arrow 1 of f of x$

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

h. $lim over x right arrow 3 of f of x$

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

l. $lim over x right arrow 2 of f of x$

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Textbook Question

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

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Textbook Question

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

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Textbook Question

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

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Textbook Question

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\left(2\right)$

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Textbook Question

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\left(4\right)$

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Textbook Question

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

${\displaystyle\lim_{x\to4}h\left(x\right)}$

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Textbook Question

Use the graph of $g (x)$ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

$\lim_{x\to2}g\left(x\right)$

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Textbook Question

Use the graph of g(x) in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

$lim over x right arrow 4 of g of x$

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Textbook Question

Use the graph of $g$ in the figure to find the following values or state that they do not exist. <IMAGE>

${\displaystyle\lim_{x\to0}g\left(x\right)}$

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Textbook Question

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(1\right)$

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Textbook Question

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(0\right)$

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Textbook Question

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>

Calculate $f\left(x\right)$ for each value of $x$ in the following table.

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Textbook Question

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>

Make a conjecture about the value of ${\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}}$ .

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Textbook Question

The function $s(t)$ represents the position of an object at time t moving along a line. Suppose $s(2)=136$ and $s(3)=156$ . Find the average velocity of the object over the interval of time $[2, 3]$ .

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Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

a. $[0, 2]$

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Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

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Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

a. s(t)=−16t^2+80t+60 at t=3

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Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

c. s(t)=40 sin 2t at t=0

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Textbook Question

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

a. Graph the position function, for 0≤t≤9.

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

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Textbook Question

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground?

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Textbook Question

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$ .

Make two tables, one showing values of $g$ for $t=8.9,8.99$ , and $8.999$ and one showing values of $g$ for $t=9.1,9.01$ , and $9.001$ .

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Textbook Question

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$ .

Make a conjecture about the value of ${\displaystyle\lim_{t\to9}\frac{t-9}{\sqrt{t}-3}}$ .

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Textbook Question

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$ . <IMAGE>

Calculate $g\left(x\right)$ for each value of $x$ in the following table.

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Textbook Question

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$ . <IMAGE>

Make a conjecture about the values of ${\displaystyle\lim_{x\to2^{-}}g\left(x\right)}$ , ${\displaystyle\lim_{x\to2^{+}}g\left(x\right)}$ , and ${\displaystyle\lim_{x\to2}g\left(x\right)}$ or state that they do not exist.

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Textbook Question

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.

$f (x) = \frac{x - 2}{\ln ∣ x - 2 ∣}$ ; $a equals 2$

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Textbook Question

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.

$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

a. The value of $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ does not exist.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

d. $\lim_{x \to 0} \sqrt{x}$ . (Hint: Graph y=√x)

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

e. $\lim_{x \to \frac{π}{2}} \cot x = 0$ . (Hint: Graph y=cot x)

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Textbook Question

Sketch the graph of a function with the given properties. You do not need to find a formula for the function.

f(2) = 1,lim x→2 f(x) = 3

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Textbook Question

Sketch the graph of a function with the given properties. You do not need to find a formula for the function.

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

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Textbook Question

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

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Textbook Question

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

a. lim x→−2^+ f(x)

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Textbook Question

Estimate the following limits using graphs or tables.

$\lim_{h \to 0} \frac{\ln (1 + h)}{h}$

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Textbook Question

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

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Textbook Question

Determine $\lim_{x \to \infty} f (x)$ and $\lim_{x \to - \infty} f (x)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{4 x}{20 x + 1}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{6 x^{2} - 9 x + 8}{3 x^{2} + 2}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{3 x^{3} - 7}{x^{4} + 5 x^{2}}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{40 x^{5} + x^{2}}{16 x^{4} - 2 x}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{1}{2 x^{4} - \sqrt{4 x^{8} - 9 x^{4}}}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{4 x^{3} + 1}{2 x^{3} + \sqrt{16 x^{6} + 1}}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{\sqrt[3]{x^{6} + 8}}{4 x^{2} + \sqrt{3 x^{4} + 1}}$

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Textbook Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = 4 x (3 x - \sqrt{9 x^{2} + 1})$

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Textbook Question

Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

$f (x) = \frac{x^{2} - 3}{x + 6}$

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Textbook Question

Complete the following steps for the given functions.

a. Find the slant asymptote of $f$ .

$f (x) = \frac{x^{2} - 2 x + 5}{3 x - 2}$

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Textbook Question

Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

$f (x) = \frac{x^{2} - 2 x + 5}{3 x - 2}$

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Textbook Question

Complete the following steps for the given functions.

a. Find the slant asymptote of $f$ .

$f (x) = \frac{4 x^{3} + 4 x^{2} + 7 x + 4}{x^{2} + 1}$

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Textbook Question

Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

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Textbook Question

Complete the following steps for the given functions.

a. Find the slant asymptote of $f$ .

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

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Textbook Question

Complete the following steps for the given functions.

b. Find the vertical asymptotes of f (if any).

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

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Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = - 3 e^{- x}$

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Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = 1 - \ln x$

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Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = \sin x$

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Textbook Question

Use an appropriate limit definition to prove the following limits.

lim x→1 (5x−2) =3;

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Textbook Question

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

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Textbook Question

Determine the following limits at infinity.

lim x→−∞ 3x^11

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

a. The graph of a function can never cross one of its horizontal asymptotes.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

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Textbook Question

If a function f represents a system that varies in time, the existence of lim $\lim_{t \to \infty} f (t)$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a bacteria culture is given by $p (t) = \frac{2500}{t + 1}$ .

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Textbook Question

If a function f represents a system that varies in time, the existence of lim ${\displaystyle\lim_{t\rightarrow\infty}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a culture of tumor cells is given by $p (t) = \frac{3500 t}{t + 1}$ .

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Textbook Question

If a function f represents a system that varies in time, the existence of lim ${\displaystyle\lim_{t\rightarrow\infty}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a colony of squirrels is given by $p (t) = \frac{1500}{3 + 2 e^{- 0.1 t}}$ .

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Textbook Question

The hyperbolic cosine function, denoted $\cosh\left(x\right)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh\left(x\right)=\frac{e^{x}+e^{-x}}{2}$ .

b. Evaluate $hyperbolic cosine of 0$ . Use symmetry and part (a) to sketch a plausible graph for $y = \cosh (x)$ .

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Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−1

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Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→−∞ cot^−1x

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Textbook Question

Determine the following limits at infinity.

lim x→ ∞ x^−6

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Textbook Question

Determine the following limits at infinity.

lim x→−∞ x^−11

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Textbook Question

Determine the following limits at infinity.

lim t→∞ (−12t^−5)

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Textbook Question

Determine the following limits at infinity.

lim x→∞ (3+10/x^2)

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Textbook Question

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

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Textbook Question

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

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Textbook Question

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

$f (- 1) = - 2$ , $f of 1 equals 2$ , $f of 0 equals 0$ , $\lim_{x \to \infty} f (x) = 1$ , $\lim_{x \to - \infty} f (x) = - 1$

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Textbook Question

Explain the meaning of lim x→−∞ f(x)=10.

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Textbook Question

Estimate lim θ→0 sin 2θ / sin θ using the graph in part (a).

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Textbook Question

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

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Textbook Question

Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure. <IMAGE>

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Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

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Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>4

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Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

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Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2+x−2 / x−1; a=1

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Textbook Question

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

lim x→−2^− f(x)

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Textbook Question

Postage rates Assume postage for sending a first-class letter in the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including each additional ounce).

a. Graph the function p=f(w) that gives the postage p for sending a letter that weighs w ounces, for 0<w≤3.5.

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Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

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Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^− tan x

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Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

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Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^− tan x

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Textbook Question

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

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Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=x^2−3x+2 / x^10−x^9

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Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

h(x)=e^x(x+1)^3

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Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

g(θ)=tan πθ/10

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Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=1/ √x sec x

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Textbook Question

Suppose x lies in the interval (1, 3) with x≠2. Find the smallest positive value of δ such that the inequality 0<|x−2|<δ is true.

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Textbook Question

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

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Textbook Question

Which one of the following intervals is not symmetric about x=5?

a.(1, 9)

b.(4, 6)

c.(3, 8)

d.(4.5, 5.5)

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Textbook Question

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 (8x+5)=13

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→2 (x^2+3x)=10

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→a (mx+b)=ma+b, for any constants a, b, and m

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→3 x^3=27

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→5 1/x^2=1/25

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Textbook Question

Determine the following limits.

Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

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Textbook Question

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

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Textbook Question

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L

d. If |x−a|<δ, then a−δ<x<a+δ.

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Textbook Question

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.

f(x)=x^1/3 <IMAGE>

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Textbook Question

Suppose $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ . Prove that $\lim_{x \to a} (f (x) - g (x)) = L - M$ .

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Textbook Question

Suppose ${\displaystyle\lim_{x\to a}}\text{ }f\left(x\right)=L$ . Prove that $\lim_{x \to a} (c (f (x)) = c L$ , where $c$ is a constant.

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Textbook Question

Use the precise definition of infinite limits to prove the following limits.

$\lim_{x \to 4} \frac{1}{{(x - 4)}^{2}} = \infty$

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Textbook Question

Use the precise definition of infinite limits to prove the following limits.

$\lim_{x \to - 1} \frac{1}{{(x + 1)}^{4}} = \infty$

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Textbook Question

Use the precise definition of infinite limits to prove the following limits.

$\lim_{x \to 0} (\frac{1}{x^{2}} + 1) = \infty$

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Textbook Question

Use the precise definition of infinite limits to prove the following limits.

$\lim_{x \to 0} (\frac{1}{x^{4}} - \sin (x)) = \infty$

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Textbook Question

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

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Textbook Question

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.

a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

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Multiple Choice

Find the limit by creating a table of values.

$\lim_{x\rarr0}-4x+2$