1. **State the problem:** We want to find the limit as $x$ approaches 0 of the expression $\left(1+4\sin x\right)^{\cot x}$. 2. **Recall the limit form:** This is a limit of the form $\lim_{x \to 0} (1 + f(x))^{g(x)}$ which often can be evaluated using the exponential and logarithm transformation: $$\lim_{x \to 0} (1 + f(x))^{g(x)} = e^{\lim_{x \to 0} f(x) g(x)}$$ if the limit inside the exponent exists. 3. **Identify $f(x)$ and $g(x)$:** Here, $f(x) = 4 \sin x$ and $g(x) = \cot x = \frac{\cos x}{\sin x}$. 4. **Evaluate the inner limit:** We need to find $$\lim_{x \to 0} 4 \sin x \cdot \cot x = \lim_{x \to 0} 4 \sin x \cdot \frac{\cos x}{\sin x}$$ 5. **Simplify the expression:** Cancel $\sin x$ in numerator and denominator: $$4 \cancel{\sin x} \cdot \frac{\cos x}{\cancel{\sin x}} = 4 \cos x$$ 6. **Evaluate the limit of the simplified expression:** $$\lim_{x \to 0} 4 \cos x = 4 \cdot \cos 0 = 4 \cdot 1 = 4$$ 7. **Write the final limit:** $$\lim_{x \to 0} \left(1 + 4 \sin x\right)^{\cot x} = e^{4}$$ **Answer:** $e^{4}$