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I have to evaluate $$\lim_{x\to 0}x^2\cos(2/x)$$ using one or more of the limit laws.

I am using the multiplication law and I am wondering if I am on the right track here?

I have split it up to: $$\left(\lim_{x\to 0}x^2\right)\left(\lim_{x\to 0}\;\cos(2/x)\right)$$ Since $\lim\limits_{x\to 0}x^2 = 0$, is the final answer $0$?

Thanks in advance!

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    No, you should use the squeeze law.2012-03-25
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    No, that's an invalid use of the limit laws. The multiplication law says: **if** $\lim\limits_{x\to a}f(x) =a$ **and** $\lim\limits_{x\to a}g(x) = b$, **then** $\lim\limits_{x\to a}f(x)g(x) = ab$. In order to be able to say the limit of the product is equal to the product of the limits, you need both limits to exist. Does $\lim\limits_{x\to 0}\cos(2/x)$ exist? If the answer is "no", then what you are trying to do is invalid.2012-03-25

3 Answers 3