# For what values of x, if any, does #f(x) = 1/((x-2)sin(pi+(8pi)/x) # have vertical asymptotes?

This function has a vertical asymptote for

A rational function has a vertical asymptote if its denominator equals zero. In your case, the denominator is

(x-2)\sin(pi+(8pi)/x)

As every product, it equals zero if and only if at least one of its factors equals zero.

So, we need to find out when

-\sin((8pi)/x)=0 \iff \sin((8pi)/x)=0

This function has an infinite number of solutions, given by

(8pi)/x=k\pi \iff 8/x=k \iff x=8/k

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The function f(x) has vertical asymptotes at x = 2 and x = 0.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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