Limit-taking: Is this valid 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.

Certainly you need that $f$ is continuous.  If not, there could be values of $f(x+\epsilon)$ very different from $f(x)$ even for very small $\epsilon$.  If $f$ is continuous this is correct.  Do you have the theorem that the limit of a product is the product of the limits?  You have $\lim\limits_{\epsilon\to0} {cf(x)f(x+\epsilon)\over c+\epsilon}=f(x)\lim\limits_{\epsilon\to0} (\frac{c}{c+\epsilon})f(x+\epsilon)$
