Is the Limit Infinity or Does It Not Exist at a Vertical Asymptote?

[Jimmy25]

This has been bugging for a while and I haven't found an answer.

Say you have a function with a vertical asymptote. This asymptote approaches infinity from both sides.

The limit approaching from either side would be infinity. So would you say the limit is infinity or does not exist?

 

[Mark44]

Here's a function that does what you describe - f(x) = 1/x².

$$\lim_{x \to 0} \frac{1}{x^2}~=~\infty$$

In one sense, the limit does not exist, because infinity is not a number in the reals. What this limit is saying is that the closer x gets to 0 (from either side), the larger 1/x² gets.

 

[HallsofIvy]

This has been bugging for a while and I haven't found an answer.

Say you have a function with a vertical asymptote. This asymptote approaches infinity from both sides.

The limit approaching from either side would be infinity. So would you say the limit is infinity or does not exist?

You can say either one. "Infinity" is not a real number so saying that a limit is "infinity" (or "negative infinity) is just saying that the limit does not exist for a particular reason.