Skip to main content
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 2.7.45
Chapter 2, Problem 2.7.45

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


limx→41(x−4)2=∞{\(\displaystyle\]\lim\)_{x\(\to\)4}}\(\frac{1}{\left(x-4\right)^2}\)=\(\infty\)

Verified step by step guidance
1
Step 1: Understand the definition of an infinite limit. The statement \( \lim_{{x \to 4}} \frac{1}{{(x-4)^2}} = \infty \) means that for every positive number \( M \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 4| < \delta \), then \( \frac{1}{{(x-4)^2}} > M \).
Step 2: Start by manipulating the inequality \( \frac{1}{{(x-4)^2}} > M \). This can be rewritten as \( (x-4)^2 < \frac{1}{M} \).
Step 3: Solve the inequality \( (x-4)^2 < \frac{1}{M} \) for \( x \). This gives \( |x-4| < \frac{1}{\sqrt{M}} \).
Step 4: Choose \( \delta = \frac{1}{\sqrt{M}} \). This choice of \( \delta \) ensures that whenever \( 0 < |x - 4| < \delta \), the inequality \( \frac{1}{{(x-4)^2}} > M \) holds true.
Step 5: Conclude that since for every \( M > 0 \), there exists a \( \delta = \frac{1}{\sqrt{M}} \) such that \( 0 < |x - 4| < \delta \) implies \( \frac{1}{{(x-4)^2}} > M \), the limit \( \lim_{{x \to 4}} \frac{1}{{(x-4)^2}} = \infty \) is proven.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a certain value, where the output grows without bound. Specifically, if the limit of a function as x approaches a value c is infinity, it indicates that the function's values increase indefinitely as x gets closer to c. This concept is crucial for understanding how functions behave near points of discontinuity or vertical asymptotes.
Recommended video:
05:50
One-Sided Limits

Limit Definition

The formal definition of a limit involves the epsilon-delta criterion, which provides a rigorous way to describe the behavior of functions as they approach a specific point. For a limit to exist, for every small positive number (epsilon), there must be a corresponding small distance (delta) such that if the input is within that distance of the point, the output is within the specified range. This definition is foundational for proving limits, especially in cases involving infinity.
Recommended video:
05:43
Definition of the Definite Integral

Continuity and Discontinuity

Continuity at a point means that a function is defined at that point, the limit exists, and the limit equals the function's value. Discontinuity occurs when any of these conditions fail, often leading to infinite limits. Understanding the types of discontinuities, such as removable or essential, is essential for analyzing limits, particularly when approaching points where the function may not be defined or behaves erratically.
Recommended video:
05:34
Intro to Continuity
Related Practice
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

Textbook Question

Evaluate each limit. 


limx→π2sin(x)−1sin(x)−1{\(\displaystyle\[\lim\)_{x\(\to\]\frac{\pi}{2}\)}{\(\frac{\sin\left(x\right)-1}{\sqrt{\sin\left(x\right)}\)-1}}}

2
views
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

Textbook Question

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.


g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

Textbook Question

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 


{0,1/2,2/3,3/4,…}, which is defined by f(n) = (n−1) / n, for n=1,2,3,…

Textbook Question

Use an appropriate limit definition to prove the following limits.


lim x→ 5x^2 − 25 / x − 5=10

1
views