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Dama28
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
\[\lim_{x \rightarrow \frac{ \Pi }{ 4 }} \frac{ \tan(x)-1 }{ x-\frac{ \Pi }{ 4 } }\]
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 : \(\huge \lim \limits_{h \rightarrow 0}\dfrac{f(a+h)-f(a)}{h}\)
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.
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.