Q. 98 - Calculus

Question

Use the quotient rule for limits and the continuity of $\sin x$ and $\cos x$ to prove that $f\left(x\right)=\tan x$ is continuous on its domain.

Step-by-Step Solution

Answer

It is proved that $f\left(x\right)=\tan x$  is continuous on its domain.

Step 1. Given Information

We are given that $\sin x$ and  $\cos x$ are continous.

Step 2. Proving the statement

Given $c\in \mathrm{ℝ}$,

$$\begin{gathered} \underset{x\to c}{\lim }f\left(x\right)=\underset{x\to c}{\lim }\tan x \\ =\underset{h\to 0}{\lim }\tan (c+h) \\ =\underset{h\to 0}{\lim }\frac{{\displaystyle \sin (c+h)}}{{\displaystyle \cos (c+h)}} \\ =\underset{h\to 0}{\lim }\left(\frac{{\displaystyle \sin c\cos h+\cos c\sin h}}{{\displaystyle \cos c\cos h-\sin c\sin h}}\right) \\ =\frac{{\displaystyle \sin c\left({\lim }_{h\to 0}\cos h\right)+\cos c\left({\lim }_{h\to 0}\sin h\right)}}{{\displaystyle \cos c\left({\lim }_{h\to 0}\cos h\right)-\sin c\left({\lim }_{h\to 0}\sin h\right)}} \\ =\frac{{\displaystyle \sin c\left(1\right)+\cos c\left(0\right)}}{{\displaystyle \cos c\left(1\right)-\sin c\left(0\right)}} \\ =\frac{{\displaystyle \sin c}}{{\displaystyle \cos c}} \\ =\tan c \\ =f\left(c\right) \end{gathered}$$

Hence $\tan x$ is continous.