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

1 Answer
Apr 10, 2016

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.