GATE Chemical 2017 | Question: 1

Question

The value of $\underset{x\rightarrow 0}{\lim}\:\frac{\tan\left ( x \right )}{x}$ is ________________.

Answer 1

\[
\lim_{x \to 0} \frac{\tan(x)}{x}
\]
Rewrite $\tan(x)$ as $\frac{\sin(x)}{\cos(x)}$:
\[
\frac{\tan(x)}{x} = \frac{\sin(x)}{x \cos(x)} = \frac{1}{\cos(x)} \cdot \frac{\sin(x)}{x}
\]
As $x \to 0$, $\cos(x) \to 1$ and $\frac{\sin(x)}{x} \to 1$, so
\[
\lim_{x \to 0} \frac{\tan(x)}{x} = 1 \times 1 = 1
\]