Provide important limits and series expansions related to trigonometric, logarithmic, and exponential functions, as well as L'Hospital's rule.

Steps to Solve

1. Identify the Types of Limits First, recognize that the problem involves evaluating different types of limits: trigonometric, logarithmic, and exponential limits, as well as applying L'Hospital's Rule.

2. Apply Trigonometric Expansions Use the provided trigonometric expansions to simplify expressions involving $\sin x$, $\tan x$, and $\cos x$. For example, using the expansion for $\sin x$: $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots $$

3. Evaluate Logarithmic Limits Refer to the logarithmic limit formulas such as: $$ \lim_{x \to 0} \frac{\log(1 + x)}{x} = 1 $$ and $$ \lim_{x \to 0} \frac{\log(1 - x)}{-x} = 1 $$

4. Evaluate Exponential Limits For exponential limits, apply the series expansions for $e^x$ and $a^x$. Use results like: $$ \lim_{x \to 0} \frac{e^x - 1}{x} = 1 $$ and $$ \lim_{x \to 0} \frac{a^x - 1}{x} = \log_e a $$

5. Apply L'Hospital's Rule where Necessary If any limit evaluates to an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, use L'Hospital's Rule: $$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} $$ as long as the limits exist.