1. Consider . Because its interval of convergence is on that same interval it is uniformly convergent. Also note that . So now since we know this series is uniformly convergent we may say this

2. Direct application of number one

3. Using the same agrument as before we can see that

3. Using the everywhere convergent Maclaurin series we may rewrite our limit as follows

4. We know that . So since if we can see that . Therefore we may rewrite our limit as