Evaluate the limit.

To evaluate this we’ll use the definition of a real number raised to a real power (i.e., ), and use the continuity of the exponential function. Then we apply L’Hopital’s rule twice. (We also use the notation to avoid huge expressions in the exponent.)

The second to last line follows since by Example #3 on page 302 of Apostol, which means the numerator is going to 0 as . However, in the denominator both the terms and go to 0 as , but then the in the denominator means the denominator is going to 1 as . Hence, the whole expression is going to 0.

Looks like an error in the 2nd application of L’Hopital, in the denominator; (xlogx+x)’ is logx+2, not logx+1. Works out to the same answer though.

Yeah, you’re right. It did still get to the right answer, but wasn’t as simple as when those terms cancelled everywhere. Anyway, I think it’s fixed now.