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

Answer 1

#x=2# and #x=ln3#

Vertical asymptotes occur when the denominator of a function equals #0#.

#(x-2)(x-2)(e^x-3)=0#

#(x-2)^2(e^x-3)=0#

Like when factoring and solving a quadratic equation, when we have multiple terms being multiplied by one another and equalling #0#, any of the terms can be equal to #0# for the expression. Thus, we can set both of the multiplied equal to #0#.

#mathbf((1))#

#(x-2)^2=0#

#x-2=0#

#x=2#

#mathbf((2))#

#e^x-3=0#

#e^x=3#

#x=ln3#

The vertical asymptotes occur at #x=2# and #x=ln3#.

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Answer 2

Scarlett Allison

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