I am trying to study a function defined as $$ f(x)=\begin{cases} x\exp\left(\dfrac{1}{x}\right); & x ≠ 0\\ 0; & x=0 \end{cases} $$ I guess the limit of $f(x)$ as $x → 0$ is undefined ($∞ × 0$) and thus the function would not be continuous at $0$.
But I would like to investigate the left and right handed limits of $f(x)$ to know which of them tends to $0$ so as to know if it is continous from the left or from the right at $0$.
Could someone show me how to investigate the left and right limits of $f(x)$ as $x → 0$?
Substituting $t=1/x$, we find that $$\lim_{x\to 0,\,x>0}\,x\exp(1/x)=\lim_{t\to+\infty}\frac{e^t}{t}=+\infty$$ and $$\lim_{x\to 0,\,x<0}\,x\exp(1/x)=\lim_{t\to-\infty}\frac{e^t}{t}=0.$$ Note that when $x<0$, $t$ approaches $-\infty$ instead of $+\infty$ as $x$ approaches $0$, since $1/x$ is negative in this case. Thus, if we restrict the domain of the function to $x\leq0$, it becomes continuous.