# For what values of x, if any, does #f(x) = 1/((x-5)sin(pi+1/x) # have vertical asymptotes?

graph{1/((x-5)sin(pi+1/x)) [-0.996, 0.996, -0.498, 0.498]}

graph{1/((x-5)sin(pi+1/x)) [-14.41, 17.46, -8.86, 7.08]}

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The function f(x) has vertical asymptotes at x = 5 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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- For what values of x, if any, does #f(x) = 1/((x-9)(x-1)(x-3)) # have vertical asymptotes?

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