Evaluate: _ x 4 - x- 4

Question

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

✓

Answer

Given that $\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\sin\text{x}-\cos\text{x}}{\text{x}-\frac{\pi}{4}}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\sqrt{2}\Big(\frac{1}{\sqrt{2}}\sin\text{x}-\frac{1}{\sqrt{2}}\cos\text{x}\Big)}{\text{x}-\frac{\pi}{4}}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\sqrt{2}\Big(\cos\frac{\pi}{4}\sin\text{x}-\sin\frac{\pi}{4}\cos\text{x}\Big)}{\text{x}-\frac{\pi}{4}}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\sqrt{2}\sin\Big(\text{x}-\frac{\pi}{4}\Big)}{\text{x}-\frac{\pi}{4}}$
$=\sqrt{2}.1=\sqrt{2}$
Hence, the required answer is $\sqrt{2}.$

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