What is a vertical asymptote in calculus?
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
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.
Stewart, James. Calculus.