Power Series and Limits
Key Questions

To be honest, I would not use power series on this one since this is a perfect problem to demonstrate the application of Squeeze Theorem. Here is how:
We know
#1 le sinx le 1# #Rightarrow 3 le 3sinx le 3# #Rightarrow 3/e^x le {3sinx}/e^x le 3/e^x# .Since
#lim_{x to infty}(3/e^x)=3/infty=0# and
#lim_{x to infty}3/e^x=3/infty=0# ,we conclude that
#lim_{x to infty}{3sinx}/e^x=0# by Squeeze Theorem.
I hope that this was helpful.

Here is a simple application of a power series in evaluating a limit.
#lim_{x to 0}{sinx]/x# by replacing
#sinx# by its Maclaurin series.#=lim_{x to 0}{xx^3/{3!}+x^5/{5!}x^7/{7!}+cdots}/{x}# by distributing the division to each term,
#=lim_{x to 0}(1x^2/{3!}+x^4/{5!}x^6/{7!}+cdots)# by sending
#x# to zero,#=10+00+cdots# since all but the first term are zero,
#=1#
I hope that this was helpful.
Questions
Power Series

Introduction to Power Series

Differentiating and Integrating Power Series

Constructing a Taylor Series

Constructing a Maclaurin Series

Lagrange Form of the Remainder Term in a Taylor Series

Determining the Radius and Interval of Convergence for a Power Series

Applications of Power Series

Power Series Representations of Functions

Power Series and Exact Values of Numerical Series

Power Series and Estimation of Integrals

Power Series and Limits

Product of Power Series

Binomial Series

Power Series Solutions of Differential Equations