Problem: Evaluate the following limit (if it exists)

Solution: We should always change roots into powers, then we will see what type we have by substituting infinity and using the limit algebra:

We see that the fraction has two parts. The denominator converges to one by the limit algebra. We can calculate its limit separately from the numerator and then put them together again, which in this case means that we simply ignore it (dividing by one means no change).

We already saw above what type this power is: ∞⁰. This is an indeterminate power, and the appropriate method calls first for using the "e to ln" trick, passing to functions and then pulling the nice exponential function out of the limit to simplify our work.

What is the type of the fraction in the limit? When we substitute infinity, we see that it is of the type infinity over infinity. The appropriate box calls for the l'Hospital rule, so we try it:

The most frequent mistake when solving this type of limit is to forget to get back to the exponential. We do remember:

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