How do you know when a function has a vertical asymptote?

Answer 1

Vertical asymptotes occur when the denominator of a rational function is zero. Since we cannot divide by zero, but we can get very close to zero on either side of it, this creates an asymptote.

There are other times such as logs when they occur, but rational functions are the ones mostly commonly seen in math classes.

So the simplest of examples would be 1/x. Since we cannot divide by 0, x cannot be 0, but it can be 1/10000000 or 1/10000000000000. It can also be -1/10000 or -1/1000000000. In other words, we can get as close to zero from either the right or the left as we want. The line x=0 forms a vertical asymptote.

Now if we make the function 1/(1-x), we have the same situation where if x=1, the denominator becomes 1-1=0. So we can get as close to 1 from the right or the left and the line x=1 forms a vertical asymptote.

So the bottom line ( pun intended) is if the denominator of a rational function becomes zero with certain values of x, say x=m, then the line x=m is a vertical asymptote.