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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 ... . –  Brian M. Scott Dec 5 '12 at 5:21
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1 Answer

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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Thanks. That works but I'm not sure if we can perform that operation on power functions... Can we? –  Lavanya Dec 5 '12 at 5:20
    
Put $ y=\frac{1}{5^x} $ and see what happens. –  Mhenni Benghorbal Dec 5 '12 at 5:22
    
Thanks to all of you. I got it now! –  Lavanya Dec 5 '12 at 5:31
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Good. Then you can write it up and post it as an answer. Then, later, you can accept it. –  Gerry Myerson Dec 5 '12 at 5:35
    
Well, now I have got another doubt.. Instead of doing it the above way, if I let it remain x -> infinity, then 5^infinity becomes infinity...so on solving I would get infinity*sin(0) which is all too confusing! And if I can't directly put infinity in place of x, why is it so? Because I'll get an infinity*0 form?? –  Lavanya Dec 5 '12 at 9:20
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