the limit represents the derivative of some function ( f ) at some number ( a ). state such an ( f ) and ( a…

the limit represents the derivative of some function ( f ) at some number ( a ). state such an ( f ) and ( a ).\n lim _{x \rightarrow pi / 4} \frac{\tan x-1}{x-pi / 4} \n( \bigcirc f(x)=\tan x, a=pi )\n( \bigcirc f(x)=\tan x, a=0 )\n( \bigcirc f(x)=\tan x, a=pi / 4 )\n( \bigcirc f(x)=\tan x, a=pi / 2 )\n( \bigcirc f(x)=\tan x, a=-pi / 4 )

the limit represents the derivative of some function ( f ) at some number ( a ). state such an ( f ) and ( a ).\n lim _{x \rightarrow pi / 4} \frac{\tan x-1}{x-pi / 4} \n( \bigcirc f(x)=\tan x, a=pi )\n( \bigcirc f(x)=\tan x, a=0 )\n( \bigcirc f(x)=\tan x, a=pi / 4 )\n( \bigcirc f(x)=\tan x, a=pi / 2 )\n( \bigcirc f(x)=\tan x, a=-pi / 4 )

Answer

Explanation:

Step1: Recall the definition of the derivative

The derivative of a function ( f(x) ) at a point ( a ) is given by ( f^{\prime}(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x - a} ).

Step2: Compare with the given limit

We are given ( \lim_{x\rightarrow\frac{\pi}{4}}\frac{\tan x - 1}{x-\frac{\pi}{4}} ). If ( f(x)=\tan x ), then ( f(a)=\tan a ). We want ( \tan a = 1 ). Since ( \tan\left(\frac{\pi}{4}\right)=1 ).

Answer:

( f(x)=\tan x), (a = \frac{\pi}{4}) (i.e., the third option ( f(x)=\tan x,a=\pi/4))