Estimating the limit of x*sin(1/x) near 0 using a graph

As $x$ approaches $0$ from either side, the factor $\sin(1/x)$ oscillates rapidly between $-1$ and $1$, but this oscillation is multiplied by $x$, which is shrinking toward $0$. This means the graph of $f(x)$ is trapped between the lines $y=x$ and $y=-x$, since $|x\sin(1/x)| \le |x|$ for all $x \neq 0$.

Looking at the graph, both the upper and lower envelope lines squeeze down to the point $(0,0)$, forcing the oscillations of $f(x)$ to flatten out and approach height $0$ near $x=0$. Approaching from the left and from the right, the function values get arbitrarily close to $0$, so the two-sided limit exists and equals $0$.