How to Solve a Plug-and-Chug Limit Problem

By Mark Ryan

A few limit problems, like plug-and-chug problems, are very easy to solve. Just plug the arrow-number into the limit function, and if the computation results in a number, that’s your answer (but see the following warning). For example,


(Don’t forget that for this method to work, the result you get after plugging in must be an ordinary number, not infinity or negative infinity or something that’s undefined.)

If you’re dealing with a function that’s continuous everywhere (like the one in this example) or a function that’s continuous over its entire domain, this method will always work. These are well-duh limit problems, and, to be perfectly frank, there’s really no point to them. The limit is simply the function value. If you’re dealing with any other type of function, this method will only sometimes work — read on…

Beware of discontinuities. The plug-and-chug method works for any type of function, including piecewise functions, unless there’s a discontinuity at the arrow-number you plug in. In that case, if you get a number after plugging in, that number is not the limit; the limit might equal some other number or it might not exist.

What happens when plugging in gives you a non-zero number over zero? If you plug the arrow-number into a limit like


and you get any number (other than zero) divided by zero — like


— then you know that the limit does not exist, in other words, the limit does not equal a finite number. (The answer might be infinity or negative infinity or just a plain old “does not exist.”)