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 2.76b
Chapter 2, Problem 2.76b

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).
f(x) = (x^4−1)/(x^2−1)

Verified step by step guidance
1
To find the vertical asymptotes, set the denominator equal to zero: \(x^2 - 1 = 0\). Solve for \(x\) to find the points where the function is undefined.
This can be factored as \((x - 1)(x + 1) = 0\). Therefore, the solutions are \(x = 1\) and \(x = -1\). These are the potential vertical asymptotes.
Check if the numerator \(x^4 - 1\) can be factored to cancel any common factors with the denominator. Factor \(x^4 - 1\) as \((x^2 - 1)(x^2 + 1)\).
The expression becomes \(\frac{(x^2 - 1)(x^2 + 1)}{x^2 - 1}\). Cancel the common factor \(x^2 - 1\), leaving \(f(x) = x^2 + 1\) for \(x \neq \pm 1\).
For \(x = 1\) and \(x = -1\), analyze \(\lim_{x \to 1^-} f(x)\), \(\lim_{x \to 1^+} f(x)\), \(\lim_{x \to -1^-} f(x)\), and \(\lim_{x \to -1^+} f(x)\) using the simplified function \(f(x) = x^2 + 1\).

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.

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value from either the left or the right. This typically happens at points where the function is undefined, often due to division by zero. Identifying vertical asymptotes involves finding the values of x that make the denominator zero while ensuring the numerator is not also zero at those points.
Recommended video:
3:40
Introduction to Cotangent Graph Example 1

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, we analyze the left-hand limit (as x approaches a from the left) and the right-hand limit (as x approaches a from the right) to determine the behavior of the function near the asymptote. This helps in understanding whether the function tends to positive or negative infinity.
Recommended video:
05:50
One-Sided Limits

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions, making it easier to analyze their behavior, such as finding asymptotes. In the given function f(x) = (x^4−1)/(x^2−1), factoring both the numerator and denominator can reveal common factors and help identify points of discontinuity. This process is crucial for determining where vertical asymptotes may exist and for simplifying the function before evaluating limits.
Recommended video:
6:04
Introduction to Polynomial Functions
Related Practice
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

369
views
Textbook Question

Determine the following limits.


b. limx→2−1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{-}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

407
views
Textbook Question

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

328
views
Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1 f(x)

259
views
Textbook Question

Complete the following steps for the given functions. 


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


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

363
views
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.

463
views