Home > Calculus > Infinite Limits and Vertical Asymptotes

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

Answer 1

Luke Alvarez

Vertical asymptotes occur at #x=0, ((2n+1)pi)/2# where #n in NN#

We have

#f(x)=1/x-tanx#

which we can write as;

#f(x)=1/x-sinx/cosx#

There will be vertical asymptotes when any part of a denominator is zero, so in this case:

For #1/x# when #x=0#
For #sinx/cosx# when #cosx=0=> x=pi/2,(3pi)/2,(5pi)/2...#

ie at #x=0, ((2n+1)pi)/2# where #n in NN#

We can see these by looking at the graph:

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Answer 2

Emma Aldridge

The function f(x) = 1/x - tan(x) has vertical asymptotes at x = 0 and at any value of x where tan(x) is undefined, which occurs when x is equal to (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.