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Let $f:\mathbf{R}\to \mathbf{R}_{>0}$ be a "nice" function on $\mathbf{R}$ with positive values. (Let's say just continuous for now, but add anything you like.)

Let $a$ be a positive integer. Suppose that $$\lim_{t\to 0^+} \frac{f(t)}{t^a}$$ exists and equals a positive real number $L$. Can we conclude that there exists some constant $C>0$ such that $f(t) \leq C t^a$ for $t$ small enough? Moreover, can we take $C=L$??

Edited question. I'm only interested in the limit coming from the right.

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Presumably $a$ is even, or we only care about limit from the right. Yes, there is such a $C$. No, sometimes we cannot take $C=L$, though for any $\epsilon>0$ we can take $C=L+\epsilon$. – André Nicolas Oct 6 '11 at 20:37
up vote 3 down vote accepted

By the definition of the limit, you have: $f(t) = (L+e(t))t^a$ with $e(t)\to 0$ when $t\to 0$. That means that $e(t)$ is bounded close to $t=0$ and hence $f(t)\leq (L+\hat e)t^\alpha$ where $\hat e$ - an upper bound for $e(t)$ in some small neighborhood of $0$. That is about right-side limit - but maybe you want to know about the both-side one?

Btw, $C$ is not necessary can be equal to $L$, for example $a=2$ and $f(t) = t^2+t^4$.

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