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For what values of x, if any, does #f(x) = 1/((x-3)(x^2-27)) # have vertical asymptotes?

Answer 1

Cameron Alexander

#x=3#,
#x=3sqrt3=5.2# and
#x=-3sqrt3=-5.2#

Vertical asymptotes of a ratio of functions are identified by zeros of the denominator.

In the given function, denominator is #(x-3)(x^2-27)# or #(x-3)(x-3sqrt3)(x+3sqrt3)#and hence three asymptotes are

#x=3#, #x=3sqrt3=5.2# and #x=-3sqrt3=-5.2#

graph{1/(x^3-3x^2-27x+81) [-10, 10, -2, 2]}

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Answer 2

Axel Alvarez

The function f(x) = 1/((x-3)(x^2-27)) has vertical asymptotes at x = 3 and x = -3.

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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.