Problem: Find the domain and limits at endpoints of its intervals for the function

Solution: This function is a general power, so in order to investigate it we have to change it into its canonical form first.

Now we look for conditions that determine the domain. The exponential can accept anything, but in its exponent we have a logarithm and that requires that cos(2x) be positive. There is a second condition, the fraction expects a non-zero denominator. We look closer at the first condition. It is an inequality featuring a trig function, so we will use some approaches from the section Solving equations and inequalities in Extra. First, the condition cos(y) > 0 has the solution

Now we substitute y = 2x and solve for x.

The second condition means that sin(x) is not zero, that is, x cannot be kπ. Unfortunately, these points fall exactly in the middle of the intervals above, so we have this ugly looking domain:

Now we need to find limits at all endpoints of intervals of this set. There are infinitely many of them, but fortunately the given function is 2π-periodic, which means that it is enough to find limits at endpoints of one period. Note that one period includes two neighboring patterns from the union above, since the period of the function is 2π, while the description of the domain above has period π. For our calculations we choose for instance those with k = 0 and k = 1. Since the function has the form of exponential, we will save time by finding first the limit of the exponent and only then put in the exponential.

We see that we actually have a limit of f at 0, it is the common one-sided limit 1. Analogously we do the limits for k = 1.

We can sketch the result like this.

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