Find the limit as x approaches zero from the right of ((sqrt((1+x)/(x^2))-(1/x))

Question

I don't even know where to start with this. Help?

P.S. If you go to the wolfphram alpha website and put copy and paste ((sqrt((1+x)/(x^2))-(1/x)) into the search bar you can see what it looks like.

Thanks!

Answer 1

as long as you're at wolframalpha, type in

limit x->0+ (sqrt((1+x)/(x^2))-(1/x))

and see that the limit is 1/2. The question is, how do you figure it?

√((1+x)/x^2) = √(1+x)/x
so,

√((1+x)/(x^2))-(1/x) = √(1+x)/x - 1/x
= (√(1+x) - 1)/x

As x->0, the fraction is 0/0, so use L'Hospital's Rule to get

1/(2√(1+x)) / 1 = 1/2

Answer 2

Oh, I forgot all about L'Hospital's Rule. Thanks!