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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 2.5.53b
Chapter 2, Problem 2.5.53b

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

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1
Identify the function given: \( f(x) = \frac{x^2 - 2x + 5}{3x - 2} \).
Recall that vertical asymptotes occur where the denominator is zero and the numerator is not zero.
Set the denominator equal to zero: \( 3x - 2 = 0 \).
Solve for \( x \) to find the potential vertical asymptote: \( x = \frac{2}{3} \).
Verify that the numerator \( x^2 - 2x + 5 \) is not zero at \( x = \frac{2}{3} \) to confirm the vertical asymptote.

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Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator approaches zero while the numerator remains non-zero. These points indicate values of x where the function is undefined, leading to the function's value approaching infinity or negative infinity. To find vertical asymptotes, set the denominator equal to zero and solve for x.
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Rational Functions

A rational function is a function represented by the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions, particularly their asymptotic behavior, is crucial for analyzing their graphs and identifying points of discontinuity.
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Finding Roots of Polynomials

Finding the roots of a polynomial involves determining the values of x that make the polynomial equal to zero. This is essential for analyzing rational functions, as the roots of the numerator indicate x-intercepts, while the roots of the denominator help identify vertical asymptotes. Techniques for finding roots include factoring, using the quadratic formula, or applying numerical methods.
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Related Practice
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)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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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.

Textbook Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2 h(x)

Textbook Question

Determine the following limits.


b. limx→2x−2x5−4x3{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{x-2}{x^5-4x^3}\)

Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


b. lim x→−2 f(x)

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Textbook Question

Determine the following limits.


b. limx→3x−3x4−9x2{\(\displaystyle\]\lim\)_{x\(\to\)3}}\(\frac{x-3}{x^4-9x^2}\)