<s>[INST] How to Evaluate $\lim_{x \to \infty} 2 + 2x\sin\left(\frac{4}{x}\right)$? Here is my limit to be evaluated

$\lim_{x \to \infty} 2 + 2x\sin\left(\frac{4}{x}\right)$=?

 [/INST] Here is a simple step by step approch
$$\begin{align}
&\quad \lim_{x \to \infty} 2 + 2x\sin\left(\frac{4}{x}\right) \\
&= \lim_{x \to \infty} 2 + 8 \frac{x}{4} \sin\left(\frac{4}{x}\right) \\
&= \lim_{x \to \infty} 2 + 8 \lim_{x \to \infty} \frac{\sin\left(\frac{4}{x}\right)}{\frac{4}{x}} \\
&= \lim_{x \to \infty} 2 + 8 \lim_{u \to 0} \frac{\sin(u)}{u} \\
&=2+8(1)=10
\end{align} 
$$
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