Question

Write the value of $\underset{x\to \infty }{\lim }\frac{\sin x}{x}.$

Answer 1

$$\begin{gathered} \underset{x\to \infty }{\lim }\left(\frac{\sin x}{x}\right) \\ \mathrm{Let}x=\frac{1}{y} \\ \mathrm{If}x\to \infty ,\mathrm{then}y\to 0. \\ =\underset{y\to 0}{\lim }y·\sin \left(\frac{1}{y}\right) \\ \mathrm{LHL}: \\ \mathrm{Let}y=0-h \\ \mathrm{If}y\to 0,\mathrm{then}h\to 0. \\ =\underset{h\to 0}{\lim }\left(\left(0-h\right)\times \sin \left(\frac{1}{0-h}\right)\right) \\ 0\times \text{The oscillating number between -1 and 1} \\ =0 \\ \mathrm{RHL}: \\ \underset{y\to 0^{+}}{\lim }\left(y·\sin \left(\frac{1}{y}\right)\right) \\ \mathrm{Let}y=0+h \\ \mathrm{If}y\to 0,\mathrm{then}h\to 0. \\ =\underset{h\to 0}{\lim }h\times \sin \left(\frac{1}{h}\right) \\ =0\times \text{The oscillating number between –1 and 1} \\ =0 \end{gathered}$$