Question

if (f(x)=sec x), then (lim_{x
ightarrow\frac{pi}{3}}\frac{f(x)-f(\frac{pi}{3})}{x - \frac{pi}{3}}) is
a 0
b (sec(\frac{pi}{3}))
c (sec(\frac{pi}{3})\tan(\frac{pi}{3}))
d nonexistent

Explanation:

Step1: Recall the definition of the derivative

The limit $\lim_{x
ightarrow a}\frac{f(x)-f(a)}{x - a}$ is the definition of the derivative of the function $y = f(x)$ at $x=a$. Here $f(x)=\sec x$ and $a = \frac{\pi}{3}$.

Step2: Find the derivative of $f(x)=\sec x$

The derivative of $y=\sec x$ is $y'=\sec x\tan x$ (using the derivative formula $\frac{d}{dx}(\sec x)=\sec x\tan x$).

Step3: Evaluate the derivative at $x = \frac{\pi}{3}$

Substitute $x=\frac{\pi}{3}$ into $y'=\sec x\tan x$. We know that $\sec\frac{\pi}{3}=2$ and $\tan\frac{\pi}{3}=\sqrt{3}$. So $y'\big|_{x = \frac{\pi}{3}}=\sec\frac{\pi}{3}\tan\frac{\pi}{3}$.

Answer:

C. $\sec(\frac{\pi}{3})\tan(\frac{\pi}{3})$