Evaluate the following limit : lim(x→π/4) (1 - tan x)/(1 - √2 sin x)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to\pi/4} \cfrac{1-tan\text x}{1-\sqrt2\,sin \text x}\)

lim(x→π/4) (1 - tan x)/(1 - √2 sin x)

Answer 1

We have \(\lim\limits_{\text x \to\pi/4} \cfrac{1-tan\text x}{1-\sqrt2\,sin \text x}\)

If x → \(\cfrac{\pi}4,\) then x - \(\cfrac{\pi}4\) = 0, let x - \(\cfrac{\pi}4\)→ y

Since,

tan(a + b) = \(\cfrac{tan\,a+tan\,b}{1-tan\,a.tan\,b}\)

sin(a + b) = sin a. cos b + cos a. sin b

By putting these , we get

Since, \(\cfrac{tan\,y}y\) = 1

Hence, \(\lim\limits_{\text x \to\pi/4} \cfrac{1-tan\text x}{1-\sqrt2\,sin \text x}\) = 2