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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 ... .2012-12-05

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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.

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    $\infty\times0$ is what is known as an indeterminate form. So are $0/0,\infty-\infty,1^{\infty},0^0$, maybe a few others I'm forgetting. If you have a formula $f(x)$, and if $\lim_{x\to Q}f(x)$ results in an indeterminate form, then you have to do some extra work to evaluate that limit. The extra work may take the form of some algebraic manipulation, or some reasoning from geometry, or (once you get to Calculus) l'Hopital's Rule, depending on the exact form of the formula $f$. You have much to look forward to.2012-12-05