Question

find the limit lim x→0 x^2 cos 1/x.

Answer 1

Solution:

Given $\varepsilon>0$ , choose $\delta=\sqrt{\varepsilon}$

Now, if $x$ is chosen so that $0<|x-0|<\delta$ ,

then

$\begin{aligned} \left|x^{2} \cos \left(\frac{1}{x}\right)-0\right| &=\left|x^{2}\right|\left|\cos \left(\frac{1}{x}\right)\right| \\ &=x^{2}\left|\cos \left(\frac{1}{x}\right)\right| \\ & \leq x^{2}(1) \\ &<(\sqrt{\varepsilon})^{2}=\varepsilon \end{aligned}$

That is: if $0<|x-0|<\delta$ , then $\left|x^{2} \cos \left(\frac{1}{x}\right)-0\right|<\varepsilon$

So, by definition of limit, $\lim _{x \rightarrow 0}\left(x^{2} \cos \left(\frac{1}{x}\right)\right)=0.$