Question

How do you determine the limit of `(x^2 -2x) / (x^2 - 4x + 4)` as x approaches 2-?

Answer 1

`lim_(x->2^-) (x^2-2x)/(x^2-4x+4) = -oo`

Explanation:

`lim_(x->2^-) (x(x-2))/((x-2)(x-2))`
`lim_(x->2^-) x/(x-2)`

If we put in values close to 2 from the left of 2 like 1.9, 1.99..etc we see that our answer gets bigger in the negative direction going to negative infinity.

`lim_(x->2^-) x/(x-2) =-oo`

If you graph it as well you will see that as x comes to 2 from the left y drops without bound going to negative infinity.

You can also use L'Hopital's Rule but it will be the same answer.