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

Answer 1

Christopher Allers

At #x=0# and at odd multiples of #pi/2#

#f(x) = 1/x-tanx = 1/x-sinx/cosx=(cosx-xsinx)/(xcosx)#

The denominator is #0# and the numerator is not at #x=0# and at odd multiples of #pi/2#.

So there are vertical asymptotes at those value of #x#.

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Answer 2

Christopher Agresta

The function f(x) = 1/x - tan(x) has vertical asymptotes at x = 0 and at the values where tan(x) is undefined, which occur at x = (n + 1/2)π, where n is an integer.

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Answer from HIX Tutor

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.