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#.