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I have a homework question to find the limit of

$\underset{x\to \infty }{\mathop{\lim }}\,\frac{2{{x}^{4}}+{{x}^{3}}+{{x}^{2}}\sin x}{{{x}^{2}}-5{{x}^{4}}+{{x}^{3}}\sqrt{x}}$

I have tried for quite a while but can't seem to figure this one out..

Could someone please help me? Thanks a-lot :)

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    Picky remark: That's no equation (since there's no "equals" sign in sight!). "Expression" or "function" would be a better word to use in this context.2011-11-18

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HINT: When finding the limit of a rational function,

$ \lim_{x\to \infty} \frac{ P(x) }{Q(x) } $

we divide through by the highest power that occurs to see the limit.

For example, $ \lim_{x\to \infty} \frac{ 2x^4 + x^3}{x^2- 5x^4} $

$= \lim_{x\to\infty} \frac{ 2+ \frac{1}{x} }{\frac{1}{x^2} - 5 } = \frac{ 2 + 0}{ 0-5} = -\frac{2}{5}.$

The reason we do this is because the highest power is the fastest increasing term, so all terms which are weaker will disappear in the limit after we divide through. The same principle applies to your limit. What do you think you should divide through by?

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    Thanks that makes it perfectly clear :)2011-11-18