`lim_(x -> pi/4) (1 - tan(x))/(sin(x) - cos(x))` Find the limit.

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Chapter 3, 3.3 - Problem 47 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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kspcr111 | In Training Educator

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`lim_(x -> pi/4) (1 - tan(x))/(sin(x) - cos(x))`

Using the L'Hopital's Rule we get

`=lim_(x -> pi/4) ((1 - tan(x))')/((sin(x) - cos(x))')`

As   `d/dx (1 - tan(x)) = -sec^2(x)`

and  `d/dx (sin(x) - cos(x))=sin(x) + cos(x)`

so,

`=lim_(x -> pi/4) ((1 - tan(x))')/((sin(x) - cos(x))')`

`= lim_(x -> pi/4) -sec^2(x) /(sin(x) + cos(x))`

`=(-sec^2(pi/4)) /(sin(pi/4) + cos(pi/4))`

`= -sqrt(2)`

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