Question

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

Answer 1

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