Nilai lim 1 - tan x per sin x - cos x , x melewati pi per 4 adalah.....
Takamori37
Anggap:
![$\begin{align}\lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{\sin x-\cos x}&=\frac{\sin\frac{\pi}{4}+\cos\frac{\pi}{4}}{1+\tan\frac{\pi}{4}}\times\lim_{x\to\frac{\pi}{4}}\frac{\frac{\cos^2x-\sin^2x}{\cos^2x}}{\sin^2x-\cos^2x} \\ &=\frac{\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{2}}{1+1}\times\lim_{x\to\frac{\pi}{4}}\frac{-(\sin^2x-\cos^2x)}{\cos^2x(\sin^2x-\cos^2x)} \\ &=\frac{\sqrt{2}}{2}\times\lim_{x\to\frac{\pi}{4}}\left[-\frac{1}{\cos^2x}\right] \\ \end{align}](https://tex.z-dn.net/?f=%24%5Cbegin%7Balign%7D%5Clim_%7Bx%5Cto%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cfrac%7B1-%5Ctan+x%7D%7B%5Csin+x-%5Ccos+x%7D%26%3D%5Cfrac%7B%5Csin%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%5Ccos%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%7B1%2B%5Ctan%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Ctimes%5Clim_%7Bx%5Cto%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cfrac%7B%5Cfrac%7B%5Ccos%5E2x-%5Csin%5E2x%7D%7B%5Ccos%5E2x%7D%7D%7B%5Csin%5E2x-%5Ccos%5E2x%7D+%5C%5C+%26%3D%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%5Csqrt%7B2%7D%2B%5Cfrac%7B1%7D%7B2%7D%5Csqrt%7B2%7D%7D%7B1%2B1%7D%5Ctimes%5Clim_%7Bx%5Cto%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cfrac%7B-%28%5Csin%5E2x-%5Ccos%5E2x%29%7D%7B%5Ccos%5E2x%28%5Csin%5E2x-%5Ccos%5E2x%29%7D+%5C%5C+%26%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5Ctimes%5Clim_%7Bx%5Cto%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cleft%5B-%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D%5Cright%5D+%5C%5C+%5Cend%7Balign%7D "$\begin{align}\lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{\sin x-\cos x}&=\frac{\sin\frac{\pi}{4}+\cos\frac{\pi}{4}}{1+\tan\frac{\pi}{4}}\times\lim_{x\to\frac{\pi}{4}}\frac{\frac{\cos^2x-\sin^2x}{\cos^2x}}{\sin^2x-\cos^2x} \\ &=\frac{\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{2}}{1+1}\times\lim_{x\to\frac{\pi}{4}}\frac{-(\sin^2x-\cos^2x)}{\cos^2x(\sin^2x-\cos^2x)} \\ &=\frac{\sqrt{2}}{2}\times\lim_{x\to\frac{\pi}{4}}\left[-\frac{1}{\cos^2x}\right] \\ \end{align}")
![$\begin{align}\lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{\sin x-\cos x}&=\frac{\sqrt{2}}{2}\times\left[-\frac{1}{\cos^2\frac{\pi}{4}}\right] \\ &=-\frac{\sqrt{2}}{2}\times\frac{1}{(\frac{1}{2}\sqrt{2})^2} \\ &=-\frac{\sqrt{2}}{2}\times\frac{1}{^1/_2} \\ &=-\frac{\sqrt{2}}{2}\times 2 \\ \lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{\sin x-\cos x}&=\boxed{-\sqrt{2}}\end{align}](https://tex.z-dn.net/?f=%24%5Cbegin%7Balign%7D%5Clim_%7Bx%5Cto%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cfrac%7B1-%5Ctan+x%7D%7B%5Csin+x-%5Ccos+x%7D%26%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5Ctimes%5Cleft%5B-%5Cfrac%7B1%7D%7B%5Ccos%5E2%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cright%5D+%5C%5C+%26%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5Ctimes%5Cfrac%7B1%7D%7B%28%5Cfrac%7B1%7D%7B2%7D%5Csqrt%7B2%7D%29%5E2%7D+%5C%5C+%26%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5Ctimes%5Cfrac%7B1%7D%7B%5E1%2F_2%7D+%5C%5C+%26%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5Ctimes+2+%5C%5C+%5Clim_%7Bx%5Cto%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cfrac%7B1-%5Ctan+x%7D%7B%5Csin+x-%5Ccos+x%7D%26%3D%5Cboxed%7B-%5Csqrt%7B2%7D%7D%5Cend%7Balign%7D "$\begin{align}\lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{\sin x-\cos x}&=\frac{\sqrt{2}}{2}\times\left[-\frac{1}{\cos^2\frac{\pi}{4}}\right] \\ &=-\frac{\sqrt{2}}{2}\times\frac{1}{(\frac{1}{2}\sqrt{2})^2} \\ &=-\frac{\sqrt{2}}{2}\times\frac{1}{^1/_2} \\ &=-\frac{\sqrt{2}}{2}\times 2 \\ \lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{\sin x-\cos x}&=\boxed{-\sqrt{2}}\end{align}")
24 votes
Thanks 56
malianda
mr crusaders bantu aku lagi ya..... nih cari msh yg mtk kok. mksih ya smoga berkah.amien
