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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.32
Chapter 2, Problem 2.32

Evaluate each limit and justify your answer. 
lim x→2 (3 / 2x^5−4x^2−50)^4

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1
Identify the type of limit problem: This is a limit of a function as x approaches a specific value.
Substitute x = 2 into the function to check if it results in an indeterminate form or a defined value.
Calculate the expression inside the parentheses: \(2x^5 - 4x^2 - 50\) by substituting x = 2.
Evaluate the expression \(3 / (2x^5 - 4x^2 - 50)\) using the result from the previous step.
Raise the result to the power of 4 to find the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating expressions that may not be directly computable at those points.
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One-Sided Limits

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to undefined expressions or different limit values.
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Intro to Continuity

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers and are combined using addition, subtraction, and multiplication. In the given limit, the polynomial in the denominator can affect the limit's value, especially if it approaches zero, leading to potential indeterminate forms.
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Introduction to Polynomial Functions
Related Practice
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)=sinxf\(\left\)(x\(\right\))=\(\sin\) x

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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

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

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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

Find all vertical asymptotes x=ax=a of the following functions. For each value of aa, determine limx→a+f(x){\(\displaystyle\]\lim\)_{x\(\to\) a^{+}}}f\(\left\)(x\(\right\)), limx→a−f(x){\(\displaystyle\]\lim\)_{x\(\to\) a^{-}}}f\(\left\)(x\(\right\)), and limx→af(x){\(\displaystyle\]\lim\)_{x\(\to\) a}}f\(\left\)(x\(\right\)).

f(x)=x+1x3−4x2+4xf\(\left\)(x\(\right\))=\(\frac{x+1}{x^3-4x^2+4x}\)

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Textbook Question

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


limx→0(1x4−sin(x))=∞{\(\displaystyle\[\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^4}\)-\(\sin\]\left\)(x\(\right\))\(\right\))=\(\infty\)

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