# What is a vertical asymptote in calculus?

##### 1 Answer

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

This is because as

On the graph of a function **limit** of the function approaches

For a more rigorous definition, James Stewart's *Calculus*,

"Definition: The line x=a is called a **vertical asymptote** of the curve

#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

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

As we approach

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

#f(x) = 1/x# has a vertical asymptote at#x=0# .

See image below.

Sources:

Stewart, James. *Calculus*.