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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 2.5.56a
Chapter 2, Problem 2.5.56a

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


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

Verified step by step guidance
1
Identify that the function \( f(x) = \frac{3x^2 - 2x + 5}{3x + 4} \) is a rational function where the degree of the numerator is one more than the degree of the denominator, indicating a slant asymptote.
Perform polynomial long division on the numerator \( 3x^2 - 2x + 5 \) by the denominator \( 3x + 4 \).
Divide the leading term of the numerator \( 3x^2 \) by the leading term of the denominator \( 3x \) to get the first term of the quotient, which is \( x \).
Multiply the entire divisor \( 3x + 4 \) by the first term of the quotient \( x \) and subtract the result from the original numerator \( 3x^2 - 2x + 5 \).
Repeat the division process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor, and the quotient obtained will be the equation of the slant asymptote.

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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, 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.
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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 divisor.
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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 limits and asymptotic behavior, particularly as x approaches infinity or specific values.
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Related Practice
Textbook Question

Determine the following limits.


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

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


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

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a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

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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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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}.


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