X- 4 lim 1 - tan x1 - -2 sin x

As sinπ /4=1/√(2), tanπ /4=1⇒ limit is of 0/0 form,applying l'opital, lim_x→π /4 sec^2x/ √(x)cos x=lim_x→π/4sec^3x =(√(2))^3=2^3/2 ...

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Standard Values of Trigonometric Ratios

x→π/ 4 lim 1 ...

Question

$l i m x \to \frac{π}{4} \frac{1 - t a n x}{1 - \sqrt{2} s i n x}$

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lim x \to \frac{π}{4} \frac{1 - tan x}{1 - \sqrt{2} sin x}

lim x \to π / 2 \frac{(1 - tan \frac{x}{2})}{(1 + tan \frac{x}{2})} \frac{(1 - sin x)}{(π - 2 x)^{3}}

f (x) = \frac{1 - t a n x}{1 - \sqrt{2} s i n x}

x \neq \frac{π}{4}

x = \frac{π}{4}

f (\frac{π}{4})

I f f (x) = \frac{1 - tan x}{1 - \sqrt{2} sin x}, f o r x \neq \frac{π}{4} i s c o n t i n o u s a t x = \frac{π}{4}, f i n d f (\frac{π}{4})

lim x \to π / 4 \frac{1 - tan x}{1 - \sqrt{2} sin x}

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