Question

How do you find the Limit of `sqrt(x^2 - 9) / (x - 3)` as x approaches 3?

Answer 1

Always try substitution first. (it won't work for this one.) When finding a limit of a fraction and in doubt, rationalize either the numerator or denominator.

Explanation:

`sqrt(x^2-9)/(x-3)`

If we rationalize the numerator, we'll be able to factor and reduce, so that looks reasonable.

`sqrt(x^2-9)/(x-3) * sqrt(x^2-9)/(sqrt(x^2-9)) = (x^2-9)/((x-3)sqrt(x^2-9))`

`= ((x-3)(x+3))/((x-3)sqrt(x^2-9))`

`= (x+3)/sqrt(x^2-9)`

Now, as `x` approaches `3`, it must be on the left. (I am assuming that we want to stay in the real numbers.)

As `xrarr3^-`, the numerator is approaching `6` and the denominator is a positive number approaching `0`.

As `xrarr3^-`, the ratio is increasing without bound.

`lim_(xrarr3^-) sqrt(x^2-9)/(x-3) = lim_(xrarr3^-) (x+3)/sqrt(x^2-9) =oo`