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Let $f$ be defined on $\mathbb{R}$. Assume $\lim_{x\to 0} \frac{x^2}{f(x)} = 3$.

What can we conclude about $f$?

  • That $\lim_{x\to 0} f(x)$ DNE

  • That $f(0) = 0$. (trying to think of an example that counters this but nothing so far)

  • Not enough info to determine $\lim_{x\to 0} f(x).$

  • Or, is it none of these.

I've considered the function $f(x) = \frac{1}{3}x^2$. And it works for 2nd claim, but I want something that makes it false.

The third claim I'm not sure...

Micah Windsor
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Jerald Tenor
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  • If $\lim_{x\to 0}f(x)$ existed, when would you be able to apply the limit laws and what would they tell you? Also... since when do we care about the value of a function at $a$ when trying to find the limit as $x\to a$? – Arturo Magidin Apr 16 '20 at 20:56

1 Answers1

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In fact : $$ \lim_{x\to 0}{f\left(x\right)}=\lim_{x\to 0}{\left(\frac{f\left(x\right)}{x^{2}}\times x^{2}\right)}=\frac{1}{3}\times 0=0 $$

And unless your $ f $ is continuous, there's nothing to say about $ f\left(0\right) \cdot $

To see that take for example : \begin{aligned} f:\mathbb{R}&\rightarrow\mathbb{R}\\ x&\mapsto\left\lbrace\begin{matrix}\frac{x^{2}}{3} &\textrm{If }x\neq 0\\ 2020&\textrm{If }x=0\end{matrix}\right.\\ \end{aligned}

CHAMSI
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