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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 2.5.53a
Chapter 2, Problem 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\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

Verified step by step guidance
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 \(y = \frac{1}{3}x + \text{(constant)}\).

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Key Concepts

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

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.
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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.
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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.
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Related Practice
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)

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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. 


a. s(t)=−16t^2+80t+60 at t=3

Textbook Question

Determine the following limits.


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

Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


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

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Textbook Question

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>

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Textbook Question

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)}}.