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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.6.42
Chapter 2, Problem 2.6.42

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→−5⁻ (3x) / (2x + 10)

Verified step by step guidance
1
Identify the type of limit: This is a one-sided limit as x approaches -5 from the left, denoted as x → -5⁻.
Analyze the denominator: As x approaches -5 from the left, the expression 2x + 10 approaches 0. Specifically, since x is slightly less than -5, 2x + 10 will be slightly negative.
Analyze the numerator: The numerator 3x approaches 3(-5) = -15 as x approaches -5.
Determine the sign of the fraction: Since the numerator approaches -15 (a negative value) and the denominator approaches a small negative value, the overall fraction (3x)/(2x + 10) approaches a positive value.
Conclude the limit: As x approaches -5 from the left, the fraction (3x)/(2x + 10) approaches positive infinity, denoted as ∞.

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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 points of interest, including points where they may not be defined. Limits can be finite or infinite, and they are essential for defining derivatives and integrals.
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One-Sided Limits

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (denoted as x→a⁻) or the right (denoted as x→a⁺). This concept is crucial when dealing with functions that may have different behaviors on either side of a point, particularly at points of discontinuity.
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Infinite Limits

An infinite limit occurs when the value of a function increases or decreases without bound as the input approaches a certain point. This is indicated by writing the limit as ∞ or −∞. Understanding infinite limits is important for analyzing vertical asymptotes and the overall behavior of functions near points where they may not be defined.
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Related Practice
Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = (2x – 1)¹/³

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Textbook Question

Theory and Examples


Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

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Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)


lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

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Textbook Question

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find


b. limx→c 2f(x)g(x)

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Textbook Question

Using the Sandwich Theorem


a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?


Give reasons for your answer.


[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

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Textbook Question

Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


g(x) = 1/(2 + (1/x))

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