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For what values of x, if any, does #f(x) = 1/((12x+3)sin(pi+(6pi)/x) # have vertical asymptotes?

Answer 1

Evelyn Agard

#x=-1/4# and #x = 6/k# for integer #k#

#12x+3=0# at #x=-1/4#

sin(pi+(6pi)/x) = 0# where

#pi+(6pi)/x = kpi# for integer #k#

And that happens where #(6pi)/x = kpi# for integer #k#

Which is at #x = 6/k# for integer #k#

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Answer 2

Evelyn Agard

The function f(x) has vertical asymptotes at values of x where the denominator of the function becomes zero. In this case, the denominator is (12x+3)sin(pi+(6pi)/x). To find the values of x that make the denominator zero, we set it equal to zero and solve for x. However, sin(pi+(6pi)/x) does not have any real zeros. Therefore, the function f(x) does not have any vertical asymptotes.

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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.