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  • <img alt="\mathrm{Find \, \, \lim _{x \rightarrow \pi / 4} \frac{1-\tan (x)^2}{\sqrt{2} \times \cos (x)-1}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BFind%20%5C%2C%20%5C%2C%

\mathrm{Find \, \, \lim _{x \rightarrow \pi / 4} \frac{1-\tan (x)^2}{\sqrt{2} \times \cos (x)-1}}

Option: 1

1


Option: 2

4


Option: 3

2


Option: 4

3


Answers (1)

best_answer

Change the variable to \mathrm{y=x-\frac{\pi}{2}}. then the limit is

                     \mathrm{ \frac{-4 \sin y \cos y}{(\cos y-\sin y)^2(\cos y-\sin y-1)} }

Now both \mathrm{\cos y}and \mathrm{(\cos y-\sin y)^2} have a limit of 1 so we are reduced to

                                      \mathrm{ \frac{-4 \sin y}{\cos y-\sin y-1} }

Now \mathrm{\frac{\sin y}{y} \rightarrow 1}so we need to find the limit of

                                      \mathrm{ \frac{-4 y}{\cos y-\sin y-1} }

Now

                   \mathrm{ \frac{\cos y-\sin y-1}{y}=y \frac{\cos y-1}{y^2}-\frac{\sin y}{y} \rightarrow-1 }
Thus the limit (assuming I have made no error) is 4 .

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Gunjita

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