Question

$\underset{x\to 0}{\lim }\frac{\tan x^0}{x}$ is equal to _______________________.

Answer 1

$$\begin{gathered} \begin{array}{c}\lim \\ x\to 0\end{array}\frac{\tan x°}{x} \\ \mathrm{since}x°=\frac{\mathrm{\pi }}{180}x \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to 0\end{array}\frac{\tan {\displaystyle \frac{\mathrm{\pi }}{180}}x}{x\times {\displaystyle \frac{\mathrm{\pi }}{180}}}\times \frac{\mathrm{\pi }}{180} \\ \mathrm{i}.\mathrm{e}.\left(\begin{array}{c}\lim \\ x\to 0\end{array}\frac{\tan {\displaystyle \frac{\mathrm{\pi }}{180}}x}{{\displaystyle \frac{\mathrm{\pi }}{180}}x}\right)\left(\begin{array}{c}\lim \\ x\to 0\end{array}\frac{\mathrm{\pi }}{180}\right) \\ \mathrm{i}.\mathrm{e}.1\times \frac{\mathrm{\pi }}{180} \\ \mathrm{i}.\mathrm{e}.\frac{\mathrm{\pi }}{180} \\ \mathrm{Hence},\begin{array}{c}\lim \\ x\to 0\end{array}\frac{\tan x°}{x}=\frac{\mathrm{\pi }}{180} \end{gathered}$$