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Is the following limit taking right? I am always confused as to when we are allowed to take the term-by-term limits then combine them as the correct full limit, sometimes term-by-term limit-taking doesn't give the right "full" limit... 

$$\lim\limits_{\epsilon\to0} {cf(x)f(x+\epsilon)\over c+\epsilon}= f^2(x)$$ perhaps I need to say that $f$ is continuous?

Thanks.

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    Is $c$ nonzero?2011-11-04
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    @J.M.: Yes, $c\neq 0$.2011-11-04
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    In most natural situations, term by term limit taking, **when** it gives an answer, gives the right answer. Note that important cases that you have been introduced to, such as $\displaystyle\lim\frac{\sin x}{x}$, are consistent with the above claim. If we take the limit of the top, the limit of the bottom, we do not get a *wrong* answer. We just get the expression $\frac{0}{0}$, which is utterly unhelpful.2011-11-04

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