Question

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

Answer 1

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

Answer 2

you there ?

Answer 3

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 :
\(\huge \lim \limits_{h \rightarrow 0}\dfrac{f(a+h)-f(a)}{h}\)

Answer 4

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.

Answer 5

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.