anonymous 3 years ago The limit represents the derivative of some function f at some number a. State such an f and a. lim (tan x − 1)/(x − π/4) (x→π/4) the answer is f(x) = tan x, a = π/4 but I don't know how to get it. :S

1. anonymous

$\lim_{x \rightarrow \frac{ \Pi }{ 4 }} \frac{ \tan(x)-1 }{ x-\frac{ \Pi }{ 4 } }$

2. hartnn

you there ?

3. hartnn

Limit definition of Derivative of a function : $$\huge \lim \limits_{h \rightarrow 0}\dfrac{f(x+h)-f(x)}{h}$$ Limit definition of Derivative of a function at x=a : $$\huge \lim \limits_{h \rightarrow 0}\dfrac{f(a+h)-f(a)}{h}$$

4. hartnn

what you have is : $$\huge \lim \limits_{x \rightarrow \pi/4}\dfrac{\tan x-1}{x- \pi/4}$$ from limit definition i need just h in the denominator, so i can substitute, h = x-pi/4 as, x -> pi/4, h ->0 also, x = pi/4+h so, we have now. $$\huge \lim \limits_{h \rightarrow 0}\dfrac{\tan (\pi/4+h)-1}{h}$$ comparing this with limit definition, f(a+h) and tan (pi/4+h) we can figure out , f(x) = tan x , and a = pi/4.

5. hartnn

also, we can confirm it by verifying, f(a) = 1, [comparing tan (pi/4+h)-1 with f(a+h)-f(a)], f(a)=f(pi/4)=tan (pi/4) = 1 is true.