Complete the following steps for the given functions. a. Find the slant asymptote of f f . f ( x ) = 3 x 2 − 2 x + 5 3 x + 4 f\left(x\right)=\frac{3x^2-2x+5}{3x+4}

Welcome back. Everyone. In this problem, we want to find the Slant Asymptote of the function G of X equals two X squared plus three, X minus four, all divided by X minus two A says it Y equals two, X minus seven by equals three, X plus seven, CY equals three, X minus seven and DY equals two X plus seven. Now, first, let's ask ourselves, what do we know about a Slant Asymptote? Well, if we let G of X, OK, be equal to P of X divided by Q of X where P is the numerator and Q is the denominator, then if we divide our function and then get a quotient as a function of X plus the remainder that quotient is going to be the function that gives us our slant a symp tote. OK. And our slant Asymptote is basically our equation that tells us what direction our Asymptote varies into. But now the slant Asymptote only exists if the degree of the numerator is one higher than that of the denominator. Now let's look at our function G of X. The degree of our numerator is two. The degree of our denominator is one. So we know that the slant Asymptote exists. So all we need to do to find it is to divide the numerator by the denominator and get the caution. So let's go ahead and do that. OK? To know we know that G FX is going to be equal to and we're gonna divide it by using long division, it's gonna be equal to two X squared plus three X minus four, divided by X minus two. Now, to get it, let's divide the first term of the numerator by the first term of the denominator X goes into two X squared two X times. OK? Now we're gonna multiply X minus two by two X. OK? And subtract it, subtract it from our numerator. Two X multiplied by X is two X squared, two X multiplied by negative two equals negative four X. And now when we subtract two, X squared minus two, X squared equals 03, X minus negative four, X equals seven X. And now let's carry down our negative four. Again, let's divide our first term here seven X by our first term in the divisor X minus two. So that would be X seven X divided by X equals seven. So it's gonna be two X. Let me write them in the proper space here. OK? So this is gonna be two eggs loss seven. Now X multiplied by seven equals seven X seven, multiplied by negative two is negative 14. And now when we subtract seven, X minus seven, X equals zero, negative four minus negative 14 equals positive 10. So what this tells us then is that our quotient? OK, let's write it out here. Our quotient is two X plus seven and our remainder is 10. Like we said, the quotient represents the function of our Slant Asymptote. So that tells us then, OK, the slant aim to it is Y equals two X plus seven. Therefore, D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

4:08 minutes

Problem 2.5.56aBriggs - 3rd Edition

Textbook Question

Complete the following steps for the given functions.

a. Find the slant asymptote of $f$ .

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

Verified step by step guidance

Identify that the function

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

3 x^{2} - 2 x + 5

by the denominator

3 x + 4

Divide the leading term of the numerator

3 x^{2}

by the leading term of the denominator

3 x

to get the first term of the quotient, which is

x

Multiply the entire divisor

3 x + 4

by the first term of the quotient

x

and subtract the result from the original numerator

3 x^{2} - 2 x + 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.

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.

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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