Substitution in the limit $x^{2}\sin(\frac{1}{x})$ where $x \to \infty$

Question

So i have a question regarding substitution. It is quite obvious I have to do the substitution $u=1/x$ to solve this limit. However, I do not understand one thing. Substitution is valid because of the theorem mentioned here Limits and substitution. Clearly if we use that theorem $\lim 1/x$ where $ x \to \infty$ is equal to $0$ not $0^+$. Therefore, by the theorem $\lim_{x \to \infty} x^2sin(1/x)=\lim_{u \to 0} sin(u)/u^2$ Where this limit does not exist since depending on which side we approach it we get negative or positive infinity. I know that for some reason we say the limit of 1/x is $0^+$ but i am not sure why. Any explanation would be appreciated.

Thank you.

Answer 1

If you have $x\to +\infty$ and you take $u=\tfrac{1}{x}$, then indeed $u\to 0$ but note that $u>0$...!

Perhaps it's more clear if you consider it the other way around: with $x=\tfrac{1}{u}$, where do you need $u$ to tend to in order for $x \to +\infty$? That's not simply $0$ because for $u \to 0^-$, you would have $x=\tfrac{1}{u}\to -\infty$. So if you want $x=\tfrac{1}{u}\to +\infty$, then you need $u\to 0^+$, the right-hand limit.

Answer 2

You want to study if possible, the limit of $\displaystyle x \mapsto x^2\sin\left(\frac{1}{x}\right)$ but where ? $+ \infty$ ? $-\infty$ ?

We'll do both.

$\bullet$ First, if $ \ x \rightarrow +\infty$, with the substitution $\displaystyle u=\frac{1}{x}$ you have $\displaystyle u \rightarrow 0^{+}$. Then $$ \frac{\sin\left(u\right)}{u^2} \underset{u \rightarrow 0^{+}}{\rightarrow}+\infty $$

$\bullet$ Now if $ \ x \rightarrow -\infty$ the same substitution leads to $$ \frac{\sin\left(u\right)}{u^2} \underset{u \rightarrow 0^{-}}{\rightarrow}-\infty $$ You could have noticed that the function $\displaystyle x \mapsto x^2\sin\left(\frac{1}{x}\right)$ is odd. Hence if it diverges to $+\infty$ when $x \rightarrow +\infty$ then it diverges to $-\infty$ when $x \rightarrow -\infty$. Then the result is not a surprise.

See the graph below to convince yourself