Complete the following steps for the given functions.

Question

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

$f (x) = \frac{x^{2} - 3}{x + 6}$

Accepted Answer

Welcome back everyone. In this problem, we want to determine the vertical Asymptote of the function H of X equals three X squared plus nine divided by five minus X A says it's X equals 10 BX equals zero, CX equals negative five and DX equals five. Now, what do we know about the vertical Asymptote? 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. Now, for our denominator five minus X, let's set the denominator equal to zero check if that value makes the numerator equal to zero. And then once it doesn't then verify the behavior of the function as it approaches the value from both sides. Now, for five minus X equals zero, if we do some transposing here, if we add X to both sides, then we get X to be equal to five. Now when X equals five, OK, then the numerator, the numerator three X squared. Well, let me write it below it here. Three X squared plus nine is going to be equal to three multiplied by five squared plus nine. Three, multiplied by five squared is three multiplied by 25 or 75 plus nine. That equals 84. So what does this tell us? Well, now we know that the numerator is not equal to zero when X equals five. OK. Now that we know that we can verify the behavior of the function as X approaches five from both the left and right. OK. So verifying the behavior, verifying H of X's behavior. OK. Then if we look at it from the left, that is the limit as X approaches five from the right of three X squared plus nine divided by five minus X. Then if X is approaching five from the left side, that is for those values that X is less than five, the denominator will approach zero from the right side since the numerator is always positive, that means the function itself is going to approach infinity. Now, when we look at it from the right hand side, OK. So the limit as X approaches five from the right of our function, eh then no as five approaches as X approaches five from the right, then the denominator approaches zero from the left since the numerator is always positive and it, and the denominator is approaching zero from the left, that means the function will approach negative infinity. OK. So that tells us then that the vertical. So let's make a note of that. Therefore, OK. Therefore, since we know that vertical asymptotes occur with the function approaches infinity or negative infinity as the input approaches a certain value, then that means since we've confirmed that the vertical Asymptote, OK. Of age of X is X equals five. Therefore, D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Complete the following steps for the given functions.

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

$f (x) = \frac{x^{2} - 3}{x + 6}$