Find the limit of $\left(1+3\sin\left(x\right)\right)^{\frac{1}{x}}$ as $x$ approaches 0

Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$

Evaluate the limit $\lim_{x\to0}\left(\frac{1}{x}\right)$ by replacing all occurrences of $x$ by $0$

An expression divided by zero tends to infinity

As by directly replacing the value to which the limit tends, we obtain an indeterminate form, we must try replacing a value close but not equal to $0$. In this case, since we are approaching $0$ from the left, let's try replacing a slightly smaller value, such as $-0.00001$ in the function within the limit:

As by directly replacing the value to which the limit tends, we obtain an indeterminate form, we must try replacing a value close but not equal to $0$. In this case, since we are approaching $0$ from the right, let's try replacing a slightly larger value, such as $0.00001$ in the function within the limit:

Once we have found both limits from the left side and from the right side, we check if they are both the same for the limit to exist. Since $\lim_{x\to c^+}f(x) \neq \lim_{x\to c^-}f(x)$, then the limit does not exist