Complete the following steps for the given functions.

Question

b. Find the vertical asymptotes of $f$ (if any).

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

Accepted Answer

Welcome back everyone. In this problem, we want to identify the vertical Asymptote of the function F of X equals X squared minus four X plus four divided by two X minus eight A says it's X equals zero, BX equals two CX equals four and DX equals eight. Now, what do we know about the vertical Asymptote of a function? Well, vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a certain value, these points often correspond to the denominator of a rational function being zero provided that the numerator does not also become zero at the same point. So to identify the vertical Asymptote, first, we need to find the value that makes the denominator of our function zero. Check if that value makes the numerator zero and if it does not, then we verify the behavior of the function as X approaches that value from both the left and the right side. So first let's check the value that makes our denominator zero. OK. So if we let two X minus eight equals zero, then two X is going to be equal to eight and thus X is going to be equal to four. OK. Now at this value is the numerator equal to zero. Well, at X equals four, then X squared minus four, X plus four is going to become four squared minus four, multiplied by four plus four. OK. Four squared is 16, four, multiplied by four equals 16 and 16 minus 16 plus four is equal to four. So in other words, our numerator is not equal to zero. OK. So since our numerator is not equal to zero, when X equals four, then if we can figure out the behavior of the function as it approaches four from both the left and right sides. In other words, if it approaches either infinity or negative infinity, then we will know that X equals four is the vertical Asymptote. So verifying, verifying the behavior of F FX. Yeah. First, if we look at the behavior of FX from the left, OK, we can try to find the limit as X approaches four from the left of X squared minus four, X plus four K divided by two, X minus eight. Now if we rewrite our function just to make it a little bit easier to read here. So if we rewrite this, if we write or numerator as the perfect square, which it is OK of X X minus two all squared. And if we factor two from our denominator making it two multiplied by X minus four, then it might be easier, easier for us to see that as X approaches four from the left side, then the denominator approaches zero from the left side as well. So since the numerator is always positive, the function will approach negative infinity. Thus, our limit from the left equals negative infinity. Now, if we look at it from the right, OK. X approach is four from the right of FFX came then. Now we might see that the as X approaches four from the right, the denominator approaches zero from the right as well. And since the numerator is always positive, that means the function will approach infinity. Therefore, that tells us then that we can believe indeed that the vertical Asymptote of the function is at X equals four. If we look back at our answer choices that tells us C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

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

Key Concepts

Vertical Asymptotes

Rational Functions

Finding Roots of Polynomials

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