Define two functions as follows:
Find a value for the constant so that the limit
is finite and non-zero. Compute the value of the limit.
First, we find the derivatives and ,
Then, we evaluate the requested limit,
Now, we would like this limit to be finite and non-zero. To find a value of that makes this happen, let’s try to apply L’Hopital’s,
So, the limit in the numerator is , which means we need the denominator to have a finite, non-zero limit (in order for the application of L’Hopital’s rule to be justified and to get a finite, non-zero limit as the problem requests). The only way this can happen is if (since if then both terms in the denominator are going to 0 as and if then the denominator is going to as ). Now, plugging in this value of and evaluating the limit we have