# Consider the function F(x)= f(x)/g(x) with g(a)equals=0. Does F necessarily have a vertical asymptote at x equals=a?

Choose the correct answer below.

A.

Yes. The function F will always have a vertical asymptote at x=a because limx right arrow a of F(x)= plus or minus infinity limx→a F(x)=±∞ whenever g(a)=0.

B.

Yes. The function F will always have a vertical asymptote at x=a because limx right F(x) does not equal plus or minus infinity limx→a F(x)≠±∞ whenever g(a)=0.

C.

No. For example, if f(x)=xplus+1, g(x)equals=x^2+2x+1, and a=−1, then the function F does not have a vertical asymptote at x =−1, even though g(−1)=0.

D.

No. For example, if f(x)=x^2−4, g(x)=x−2, and a=2, then the function F does not have a vertical asymptote at x=2, even though g(2)=0.

### 2 Answers

- TomVLv 78 months agoFavorite Answer
No. If f(a) = 0, then F(a) = f(a)/g(a) = 0/0 which is indeterminate. F(a) may have a vertical asymptote at x = a, but it may simply have a hole at x = a. This is the situation for choice D and shows that a rational function does not necessarily have a vertical asymptote even though the denominator evaluates to 0.

Choice C has a correct conclusion, F does not necessarily have a vertical asymptote, but the argument is invalid because although F(x) = (x+1)/(x²+2x+1) evaluates to 0/0 at x = -1, it does have a vertical asymptote at x = -1.

The correct answer D has the correct conclusion and supports that conclusion with a valid example.

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