Question

What is a vertical asymptote in calculus?

Answer 1

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.

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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:

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

See image below.

Sources:
Stewart, James. Calculus. `6^(th)` ed. Belmont: Thomson Higher Education, 2008. Print.