# Limits - trigonometry - tending to infinity

How do we solve:

$$\lim_{x\to \infty} 5^x \sin\left(\frac{a}{5^x}\right)$$

Thank You.

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Gerry Myerson’s answer is the way to go, but you can easily see what the limit has to be if you remember that $\sin x\approx x$ when $|x|$ is small. Thus, $\sin\frac{a}{5^x}\approx\frac{a}{5^x}$ when $x$ is large, and ... . –  Brian M. Scott Dec 5 '12 at 5:21

Convince yourself that it's the same as evaluating $\lim_{t\to0}{\sin at\over t}$, and then use other stuff you know to do that one.
Put $y=\frac{1}{5^x}$ and see what happens. –  Mhenni Benghorbal Dec 5 '12 at 5:22