Evaluate the following limit: _ x 4 - ( 4-x )( + )

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{\cos\text{x}-\sin\text{x}}{\big(\frac\pi4-\text{x}\big)(\cos\text{x}+\sin\text{x})}$

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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{\cos\text{x}-\sin\text{x}}{\big(\frac\pi4-\text{x}\big)(\cos\text{x}+\sin\text{x})}$$\Rightarrow\text{x}\rightarrow\frac{\pi}{4},$ then $\frac\pi4-\text{x}\rightarrow0$ let $\frac\pi4-\text{x}=\text{y}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\cos\big(\frac\pi4+\text{y}\big)-\sin\big(\frac\pi4+\text{y}\big)}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\Big[\big(\cos\frac\pi4\cos\text{y}-\sin\frac\pi4\sin\text{y}\big)-\big(\sin\frac\pi4\cos\text{y}+\cos\frac\pi4\sin\text{y}\big)\Big]}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\Big[\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}-\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}\Big]}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\frac{-2\sin\text{y}}{\sqrt{2}}}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$
$=\sqrt{2}\lim\limits_{\text{y}\rightarrow{0}}\Big(\frac{\sin\text{y}}{\text{y}}\Big)\times\frac{1}{\lim\limits_{\text{y}\rightarrow{0}}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$
$=\sqrt{2}\times1\times\frac{1}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=\sqrt{2}\times\frac{1}{\frac{2}{\sqrt{2}}}$
$=\frac{\sqrt{2}\times\sqrt{2}}{2}=1$