Question

How do you find the limit of `(sqrt(x) - sqrt(6) + sqrt(x-6))/sqrt(x^2 - 36)` as x approaches 6+?

Answer 1

The limit is `1/sqrt12`.

Explanation:

I do not have an algebraic solution, but here is a solution method:

Split the ratio:

`(sqrt(x) - sqrt(6) + sqrt(x-6))/sqrt(x^2 - 36)`

`= (sqrt(x) - sqrt(6))/sqrt(x^2 - 36)+(sqrt(x-6))/sqrt(x^2 - 36)`

The second ratio can be written: `(cancelsqrt(x-6))/(cancelsqrt(x-6)sqrt(x+6))`. So the limit is `1/sqrt12`

The limit of `(sqrt(x) - sqrt(6))/sqrt(x^2 - 36)` can be found by l'Hospital's rule to be `0`.