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 the fraction inside be positive. There is a second condition, the fraction expects a non-zero denominator. We look closer at the first condition. The fraction should be greater than zero, which is a sign inequality and we can solve it by considering regions of signs. Dividing points are −1 and 1, we get
Thus the domain is given by the outer intervals, sharp inequality means that they should be open, which also takes care of the condition originating from the fraction (there will not be zero in the denominator).
We have four limits to evaluate. We take it from the left, for tips on limit evaluation see e.g. Limits in Functions - Methods Survey.
To save time and space we will in subsequent calculation skip the exponential, but we have to remember to return to it at the end. First, we have to find out what is the limit inside the logarithm. This is actually simple, for a ratio of two polynomials we can use the trick with factoring out leading powers, here we can actually cancel since they are the same.
We have an indeterminate product and we use the appropriate trick.
Now we have to remember to get back to the exponential.
Now we look at the limit at −1 from the left, we start without the exponential.
Therefore
The limit at 1 from the right is similar.
Finally we are getting to the limit at infinity, the calculations here are analogous to those at negative infinity.
Again, we can try to express these answers in a sketch. To save room we will change the scale on the y-axis, from the canonical form we also see that f is always positive and so we do not have to worry about negative values in our picture.
