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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 2 - Limits문제 2.5.53b
2장, 문제 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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
3:40
Introduction to Cotangent Graph Example 1

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.
추천 영상:
6:04
Intro to Rational Functions

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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6:04
Introduction to Polynomial Functions
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