Expert: Steve Holleran - 10/29/2007

Question
Question in a first college Calculus course from a chapter on L'Hopital's rule:
limit x -> infinity ((x^2 + x)^.5-x). Multiplying by ((x^2+x)^.5+x)/((x^2+x)^.5+x) makes this indiscrinate form infinity/infinity. But, taking the separate derivatives of numerator and demoninator repeatedly remains in infinity/infinity form and doesn't seem to terminate.

Answer
Hi Tom,

How about this?

If we do your first step, we have

lim(h->0) [x^2 / (sqrt(x^2 + x)+ x)]

If you take the expression in the square root, x^2 + x,

we can "factor out" an x^2 :  x^2(1 + x/x^2) = x^2(1 + 1/x)

Then you have:

           x^2 / (x*sqrt(1+1/x) + x)

=           x^2 / x * [sqrt(1 + 1/x) + 1]

So cancel an x and get:

    lim(x->inf) [x / [sqrt(1 + 1/x) + 1]

Now, as x-> inf, the 1/x in the bottom ->0, so the bottom --> 2 and you have inf/2 = inf.

Don't know quite what else to do with this one.

Steve