Evaluate the following limit: _ x 4 - x- 4

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{\sqrt{\cos\text{x}}-\sqrt{\sin\text{x}}}{\text{x}-\frac{\pi}{4}}$

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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{\sqrt{\cos\text{x}}-\sqrt{\sin\text{x}}}{\text{x}-\frac{\pi}{4}}$$=\lim\limits_{{\text{x}-{\frac\pi4\rightarrow0}}}\frac{\big(\sqrt{\cos\text{x}}-\sqrt{\sin\text{x}}\big)}{\big(\text{x}-\frac{\pi}{4}\big)}\times\frac{\big(\sqrt{\cos\text{x}}+\sqrt{\sin\text{x}}\big)}{\big(\sqrt{\cos\text{x}}+\sqrt{\sin\text{x}}\big)}$
$=\lim\limits_{{\text{x}-{\frac\pi4\rightarrow0}}}\frac{\big(\sqrt{\cos\text{x}}-\sqrt{\sin\text{x}}\big)}{\big(\text{x}-\frac{\pi}{4}\big)\big(\sqrt{\cos\text{x}}+\sqrt{\sin\text{x}}\big)}$
As $\text{x}\rightarrow\frac{\pi}{4}\Rightarrow\text{x}-\frac\pi4\rightarrow0\Rightarrow$let $\text{x}-\frac\pi4=\text{y}$
$=\lim\limits_{\text{y}\rightarrow0}\frac{\Big(\cos\big(\frac\pi4+\text{y}\big)-\sin\big(\frac\pi4+\text{y}\big)\Big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$
$=\lim\limits_{\text{y}\rightarrow0}\frac{\big(\cos\frac\pi4\cos\text{y}-\sin\frac{\pi}{4}\sin\text{y}\big)-\big(\sin\frac\pi4\cos\text{y}-\cos\frac{\pi}{4}\sin\text{y}\big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$
$=\lim\limits_{\text{y}\rightarrow0}\frac{\Big(\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}-\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}\Big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$
$=\lim\limits_{\text{y}\rightarrow0}\frac{\Big(-2\frac{\sin\text{y}}{\sqrt{2}}\Big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$
$=-\sqrt{2}\Big(\lim\limits_{\text{y}\rightarrow0}\frac{\sin\text{y}}{\text{y}}\Big)\times\frac{1}{\lim\limits_{\text{y}\rightarrow0}\sqrt{\cos\big(\text{y}+\frac\pi4}\big)+\lim\limits_{\text{y}\rightarrow0}\sqrt{\sin\big(\text{y}+\frac{\pi}{4}\big)}}$
$=-\sqrt{2}\times1\times\frac{1}{\sqrt{\cos\frac{\pi}{4}}+\sqrt{\sin\frac\pi4}}$
$=-\sqrt{2}\times\frac{1}{\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}+\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}}$ $\Big[\because\cos\frac\pi4=\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}\Big]$
$=\frac{-\sqrt{2}}{\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}+(1+1)^{\frac{1}{2}}}$
$=\frac{-\sqrt{2}}{\sqrt{2}\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}}$
$=-\frac{1}{2^{\frac14}}$

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