Question

lim as x approaches pi/4:
[(tan x - 1)/(x - pi/4)]

The limit represents the derivative of some function f at some number a. State such an f and a.

Answer 1

(tan pi/4 - 1) / (pi/4 - pi/4) = 0/0 .. Use L' Hopital's Rule and you get?..

Answer 2

whats the solution ?

Answer 3

just get the derivative of numerator and denominator... if you use LHR (L' Hopital's) , F/f will become F'/f'..

Answer 4

i think the hint is to "state" the function f(x) such that \[\large f \prime (\pi/4)=\lim_{x \rightarrow \pi/4}\frac{ \tan(x)-1 }{ x-\pi/4 }\] then compute \[\large f \prime (\pi/4)\] using THE function f(x)

Answer 5

\[\large f \prime (\pi/4)=\lim_{x \rightarrow \pi/4}\frac{\tan x - 1}{x-\pi/4}=\lim_{x \rightarrow \pi/4}\frac{\tan x - \tan(\pi/4)}{x-\pi/4}\]so the function f(x) is \[\large f(x) = \tan(x)\]

Answer 6

at the end it will be like this: lim x->pi/4 (1/cos^2(x)) right?

Answer 7

yup.

Answer 8

jeez thanks alot

Answer 9

i assume you were given \[\large f \prime (x) = \frac{ 1 }{ \cos^2x }\] when f(x) = tan x

Answer 10

nah i looked everything up at the internet.... my teacher didn't give me anything and i'm not very good at this he just said compute the limit and nothing else.

Answer 11

i see. i'll show you my solution. \[\large \frac{ \tan x - \tan(\pi/4) }{ x-\pi/4 }\frac{ 1+\tan x \tan (\pi/4) }{ 1+\tan x \tan (\pi/4) }\]

Answer 12

then switch the two denominators\[\large = \frac{ \tan x - \tan (\pi / 4) }{ 1+\tan x \tan (\pi/4) }\frac{ 1+\tan x \tan (\pi/4) }{ x- \pi/4 }\]

Answer 13

the first factor is a trigonometric identity \[\large = \tan(x-\pi/4)\frac{ 1+\tan x an (\pi/4) }{ x-\pi/4 }\]

Answer 14

use the sin/cos equivalent of tan
\[\large \frac{ \sin (x-\pi/4) }{ \cos(x-\pi/4) }\frac{ 1+\tan x \tan \pi/4 }{ x-\pi/4 }\]

Answer 15

switch denominators again, and replace tan(pi/4) by 1
\[\large= \frac{ \sin(x-\pi/4) }{ (x-\pi/4) }\frac{ 1+tanx }{ \cos(x-\pi/4) }\]

Answer 16

evaluating lim{x -> pi/4}
\[\large \lim_{x \rightarrow \pi/4} \frac{ \sin(x-\pi/4) }{ (x-\pi/4) }=1\]

Answer 17

so the problem is reduced to \[\large \lim_{x \rightarrow \pi/4}\frac {1+\tan x}{\cos (x-\pi/4)}\]
the normal substitution applies

Answer 18

wow thanks dude i'll try to figure it out