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my calculus teacher told us to not solve this problem but I gave it a try, and I failed miserably as the $\lim_{x\to ∞}$ for the sin is ∞. I'm supposed to bring the formula to a L'hopital expression at some point

Here's the problem: $$\lim\limits_{x \to ∞} (1+\sin(4x))^{\cot(x)}$$

I've tried applying natural logarithm in order to take the cot as a factor, but then I still have the issue that goes to infinity and beyond:

$$\lim\limits_{x \to ∞} e^{ln(1+\sin(4x))^{\cot(x)}}$$ so then: $$\lim\limits_{x \to ∞} e^{ln(1+\sin(4x))* \color{red}{\cot(x)}}$$

After this point I tried converting the $\color{red}{\cot(x)}$ to $\color{red}{\frac{\cos(x)}{\sin(x)}}$ and tried to operate from there, but I got really stuck due to the $\lim_{x\to ∞}$.

Anyways, I'll leave this here, I hope to find the answer myself before but I might get lucky with someone here.