Evaluate the below given limit.

Solution

Note as well that we actually do have to do the right-hand limit here. We know that the natural logarithm is just described for positive x and thus it is the only limit that makes any sense.

Now, in the limit, we obtain the indeterminate form (0) (-∞).  L'Hospital's Rule won't apply on products, it works on quotients only.  Though, we can turn it into a fraction if we rewrite things a little.

The function is the similar, just rewritten, & now the limit is in the form -∞ /∞ and now we can utilizes L'Hospital's Rule.

Now, it is a mess, however it cleans up nicely.

Now let's take a look at the indeterminate forms,

                1^∞                0⁰         ∞⁰

These can all be dealt with in the given way therefore we'll just work one example.