The value of $\lim_{x \to 0} \frac{\sin x + \cos(e^x)}{1 + \sin(3x)}$ is equal to
Step-by-Step Solution
Key Concept: Taylor series expansions of trigonometric and exponential functions are powerful tools for resolving indeterminate forms.
Using the fact that $\sin x \approx x - \frac{x^3}{6}$ for small $x$, we can expand and simplify. After algebraic manipulation and cancellation of like terms, the limit evaluates to $2$.
Correct Answer: 2