Figure 1.5.4(a) has a graph of $\frac{\sin \left(x\right)\left(x-1\right)}{x^2\+x\+1}$ in red, and graphs of $\pm \left|\frac{x-1}{x^2\+x\+1}\right|$ in blue and green, respectively. The blue and green curves are envelopes for the decreasing oscillations of $\sin \left(x\right)$. The figure suggests that $\underset{x\to \infty }{lim}\sin \left(x\right)g\left(x\right)\=0$.

Unfortunately, since $\underset{x\to \infty }{lim}\sin \left(x\right)$ does not exist, the Product rule for limits cannot be applied. Indeed, since $\left|\sin \left(x\right)\right|\le 1$, Principle 1.1.1 could be invoked. Because $\underset{x\to \infty }{lim}g\left(x\right)\=0$, it immediately follows that $\underset{x\to \infty }{lim}\sin \left(x\right)g\left(x\right)\=0$.

Expression palette: Limit operator
A space or explicit multiplication between factors in the numerator is essential.

Context Panel: Evaluate and Display Inline