Limits

and both have period . Their combination also repeats every , so period of is .

Using standard limit identity:

Let us evaluate the numerator and denominator around :

As , grows slower than any power function . Hence,

Let’s compute the numerator and denominator separately as :

Let , then:

We are required to evaluate the limit: This is an indeterminate form of the type , so we can apply logarithms to simplify: As , we observe that and , so applying L'Hôpital's rule to the limit of gives: Thus, , so .

We use series expansions of trigonometric functions around 0: Substituting these approximations into the expression: