Question

How do you find the limit of `(x^2-9)/(x^2+2x-3)` as x approaches `1^+`?

Answer 1

`lim_(xrarr1^-)(x^2-9)/(x^2+2x-3) = -oo`

Explanation:

When we try to evaluate `lim_(xrarr1^-)(x^2-9)/(x^2+2x-3)` by substitution we get a non-`0` number over `0`. This tells us that we have some kind of infinite limits at `1`.

To determine whether the ratio is increasing or decreasing without bound (going to `oo` or `-oo`) we need to investigate the signs of the factors.

`lim_(xrarr1^-)(x^2-9)/(x^2+2x-3) = lim_(xrarr1^-)(x^2-9)/((x+3)(x-1))`

As `xrarr1^+`,

`x^2-9 rarr -8`, `" "` (The numerator is negative.)
`x+3rarr4`, `" "` (This factor is positive.)
`x-1rarr0^+`, `" "` (This factor is positive.)

The ratio is decreasing without bound.

We write:

`lim_(xrarr1^-)(x^2-9)/(x^2+2x-3) = -oo`