how to find vertical asymptotes using limits

The vertical asymptote is a place where the function is undefined and the limit of the function does not exist.

This is because as #1# approaches the asymptote, even small shifts in the #x#-value lead to arbitrarily large fluctuations in the value of the function.

On the graph of a function #f(x)#, a vertical asymptote occurs at a point #P=(x_0,y_0)# if the limit of the function approaches #oo# or #-oo# as #x->x_0#.

For a more rigorous definition, James Stewart's Calculus, #6^(th)# edition, gives us the following:

"Definition: The line x=a is called a vertical asymptote of the curve #y=f(x)# if at least one of the following statements is true:

#lim_(x->a)f(x) = oo#
#lim_(x->a)f(x) = -oo#
#lim_(x->a^+)f(x) = oo#
#lim_(x->a^+)f(x) = -oo#
#lim_(x->a^-)f(x) = oo#
#lim_(x->a^-)f(x) = -oo#"

In the above definition, the superscript + denotes the right-hand limit of #f(x)# as #x->a#, and the superscript denotes the left-hand limit.

Regarding other aspects of calculus, in general, one cannot differentiate a function at its vertical asymptote (even if the function may be differentiable over a smaller domain), nor can one integrate at this vertical asymptote, because the function is not continuous there.

As an example, consider the function #f(x) = 1/x#.

As we approach #x=0# from the left or the right, #f(x)# becomes arbitrarily negative or arbitrarily positive respectively.

In this case, two of our statements from the definition are true: specifically, the third and the sixth. Therefore, we say that: