Skip to main content
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 2.5.53a
2장, 문제 2.5.53a

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


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

검증된 단계별 안내
1
Perform polynomial long division of the numerator \(x^2 - 2x + 5\) by the denominator \(3x - 2\).
Divide the leading term of the numerator \(x^2\) by the leading term of the denominator \(3x\) to get the first term of the quotient, \(\frac{1}{3}x\).
Multiply the entire divisor \(3x - 2\) by \(\frac{1}{3}x\) and subtract the result from the original numerator \(x^2 - 2x + 5\).
Repeat the process with the new polynomial obtained after subtraction to find the next term of the quotient.
The slant asymptote is the linear part of the quotient obtained from the division, which is \(y = \frac{1}{3}x + \text{(constant)}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m
도움이 되었나요?

주요 개념

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

Slant Asymptote

A slant (or oblique) asymptote occurs when the degree of the numerator of a rational function is exactly one higher than the degree of the denominator. To find it, you perform polynomial long division on the function. The quotient (ignoring the remainder) gives the equation of the slant asymptote, which describes the behavior of the function as x approaches infinity or negative infinity.
추천 영상:
5:37
Introduction to Cotangent Graph

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves dividing the leading term of the numerator by the leading term of the denominator, multiplying the entire denominator by this result, and subtracting it from the numerator. This process is repeated until the degree of the remainder is less than that of the denominator, allowing us to find the slant asymptote.
추천 영상:
6:04
Introduction to Polynomial Functions

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit various behaviors, including vertical and horizontal asymptotes, depending on the degrees of the numerator and denominator. Understanding the properties of rational functions is crucial for analyzing their graphs and determining their asymptotic behavior, including the identification of slant asymptotes.
추천 영상:
6:04
Intro to Rational Functions
관련 실천
교과서 질문

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)

397
views
교과서 질문

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. 


a. s(t)=−16t^2+80t+60 at t=3

427
views
교과서 질문

Determine the following limits.


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

379
views
교과서 질문

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

392
views
교과서 질문

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.


f(x)=x1/3 <IMAGE>

362
views
교과서 질문

The hyperbolic cosine function, denoted cosh(x)\(\cosh\]\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


a. Determine its end behavior by analyzing limx→∞cosh(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{\(\cosh\)(x)}} and limx→−∞cosh(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{\(\cosh\)(x)}}.

503
views