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Question

Question: \[\lim_{\alpha \rightarrow \pi/4}\frac{\sin\alpha - \cos\alpha}{\alpha - \pi/4} =\]...

lim⁡α→π/4sin⁡α−cos⁡αα−π/4=\lim_{\alpha \rightarrow \pi/4}\frac{\sin\alpha - \cos\alpha}{\alpha - \pi/4} =

A

2\sqrt{2}

B

1/21/\sqrt{2}

C

1

D

None of these

Answer

2\sqrt{2}

Explanation

Solution

lim⁡α→π/4sin⁡α−cos⁡αα−π/4(00form)\lim_{\alpha \rightarrow \pi/4}\frac{\sin\alpha - \cos\alpha}{\alpha - \pi/4}(\frac{0}{0}\text{form}) =lim⁡α→π/4cos⁡α+sin⁡α1\lim_{\alpha \rightarrow \pi/4}\frac{\cos\alpha + \sin\alpha}{1}

(By ‘L’ Hospital rule) = 12+12=22=2\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}.