12.4.1I'Hôpital ... meet Taylor

Your diabolical teacher has another competition. The team that can evaluate the limit below will get a gift certificate to Pizza Pi restaurant… of course, you may not use a calculator to evaluate it.

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ^ { 4 } ) - x ^ { 4 } } { x ^ { 12 } }$

 Applying l’Hôpital’s Rule to the limit in the previous problem can be time consuming, but Aurora has a clever idea. She rewrites $\sin(x^4)$ as a Taylor series centered at $x = 0$, and evaluates the limit again. Give her strategy a try. Then compare your result the result your obtained using l’Hôpital’s Rule.

Aurora wonders if the strategy she used in problem 12-112 can be used to demonstrate that l’Hôpital’s Rule holds true.

That is, she wants to show that for functions $f$ and $g$ whose derivatives exist for all orders at $x = a$, if $\lim\limits _ { x \rightarrow a } \frac { f ( x ) } { g ( x ) }$ is indeterminate such that $\lim\limits _ { x \rightarrow a } f ( x ) = 0$ and $\lim\limits _ { x \rightarrow a } g ( x ) = 0$ then $\lim\limits _ { x \rightarrow a } \frac { f ( x ) } { g ( x ) } = \lim\limits _ { x \rightarrow a } \frac { f ^ { \prime } ( x ) } { g ^ { \prime } ( x ) }$.

1. Using sigma notation, Aurora writes a generic Taylor series centered at $x = a$ for $f$ and for $g$: $f ( x ) = \displaystyle\sum _ { n = 0 } ^ { \infty } b _ { n } ( x - a ) ^ { n }$ and $g ( x ) = \displaystyle\sum _ { n = 0 } ^ { \infty } c _ { n } ( x - a ) ^ { n }$.

   1. Write the Taylor series for $f$ and for $g$ in expanded form, showing the first three terms and the general term.

   2. Explain the significance of $b_{n}$ and $c_{n}$. For example, in the context of this problem, what are $b_{0}$ and $c_{0}$? Likewise, what do $b_{1}$ and $c_{1}$ represent? 

2. Use the expanded forms to write a new expression for $\lim\limits _ { x \rightarrow a } \frac { f ( x ) } { g ( x ) }$. Then substitute values for $b_{0}$ and $c_{0}$_, factor, and evaluate.

3. Interpret your result in the context of this problem. That is, explain how this result demonstrates that $\lim\limits _ { x \rightarrow a } \frac { f ( x ) } { g ( x ) } = \lim\limits _ { x \rightarrow a } \frac { f ^ { \prime } ( x ) } { g ^ { \prime } ( x ) }$.

 Use Taylor series to evaluate the limits below.