Question

\[\lim_{x \to \infty} \frac{\sin x}{x} .\] 

Answer 1

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