Solutions to homework problems.
Remarks.
Usually, applying known results, can simplify the computation. For instance, from
, we deduce that 
Using the second limit equation, we can solve problem 378 as above. One can directly use the following results in the future too.
Sometimes, it is not hard to show that one should use L’Hôpital’s rule. For example, as in problem 383, 391 (by inverting x to 1/(1/x)). But then while applying L’Hôpital’s rule, one must take derivative for functions such as
which is a little complicated. The trick is to set 
And as 
, then 
, and vice versa. Making this substitution first, then apply L’Hôpital’s rule. The limit
in problem 386 has the form 
, which is an indefinite form. We cannot “reduce to the common denominator” in this problem because there is no denominator. To evaluate this limit, we use the known result 
, so that the original limit is equal to 
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