_ 4 - - 4 = - Vidyadip

MCQ

$\lim _{\alpha \rightarrow \frac{\pi}{4}} \frac{\sin \alpha-\cos \alpha}{\alpha-\frac{\pi}{4}}=$

✓
$\sqrt{2}$
B
$\frac{1}{\sqrt{2}}$
C
1
D
None of these

✓

Answer

Correct option: A.

$\sqrt{2}$

(A)
$\lim _{\alpha \rightarrow \frac{\pi}{4}} \frac{\sin \alpha-\cos \alpha}{\alpha-\frac{\pi}{4}}$
$=\lim _{\alpha \rightarrow \frac{\pi}{4}}\left\{\frac{\sqrt{2}\left(\sin \alpha \cdot \frac{1}{\sqrt{2}}-\cos \alpha \cdot \frac{1}{\sqrt{2}}\right)}{\left(\alpha-\frac{\pi}{4}\right)}\right\}$
$=\sqrt{2} \lim _{\alpha \rightarrow \frac{\pi}{4}} \frac{\sin \left(\alpha-\frac{\pi}{4}\right)}{\left(\alpha-\frac{\pi}{4}\right)}=\sqrt{2}(1)=\sqrt{2}$

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