Q. 99 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

We are given that $\cos x$ is continous. 

Step 2. Proving the statement

If $x=c$ is in the domain of $\sec x$, then c is not a multiple of $\frac{\pi }{2}$, and $\cos c\ne 0$. Therefore, $\underset{x\to c}{\lim }\sec x=\underset{x\to c}{\lim }\frac{1}{\cos x}$ is the quotient of $\underset{x\to c}{\lim }1$ by,

$\underset{x\to c}{\lim }\cos x=\cos c$

or

$\frac{1}{\cos c}=\sec c$