How would you solve lim(x-5)[(x-5)/((√x-1)-2)]?

Question

How would you solve                                lim(x->5)[(x-5)/((√x-1)-2)]?

anonymous · Answer

Trying to solve this directly you get lim(x -> 5)(f(x)) = 0/0 therefore, you can apply L'Hopital's rule. Taking the derivative of the top and the bottom, you get lim(x -> 5)(f(x)) = lim(x -> 5)(1/(1/(2*sqrt(x-1)))) = lim(x -> 5)(2*sqrt(x-1)). Putting x into this equation, you get lim(x -> 5)(f(x)) = 4

anonymous · Answer

How does the bottom equal 1/(2xsqrt(x-1))?

If the bottom is (sqrt(x-1) -2, how does a 2 get in front of the sqrt? I got 1/2(x-1)^(-1/2)

anonymous · Answer

Exactly, that is in the denominator 1/(1/2*(x-1)^(-1/2)) so move it all to the top canceling the 1's and you get 2*sqrt(x-1)

anonymous · Answer

I finally figured out how that makes sense! My head just exploded but its all good. Thank you :)