Final Answer
Step-by-step Solution
Specify the solving method
Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$
Multiplying the fraction by $\ln\left(1+3\sin\left(x\right)\right)$
Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$
The limit of a constant is just the constant
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
After deriving both the numerator and denominator, the limit results in
Evaluate the limit $\lim_{x\to0}\left(\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}\right)$ by replacing all occurrences of $x$ by $0$
The sine of $0$ equals
Add the values $1$ and $0$
The cosine of $0$ equals
Divide $3$ by $1$
Calculate the power $e^{3}$