PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos x}\]

\[\lim_{x \to 0} \frac{x^2 + 1 - \cos x}{x \sin x}\] 

\[\lim_{x \to 0} \frac{\sin 2x \left( \cos 3x - \cos x \right)}{x^3}\] 

\[\lim_{x \to 0} \frac{2 \sin x^\circ - \sin 2 x^\circ}{x^3}\] 

\[\lim_{x \to 0} \frac{x^3 \cot x}{1 - \cos x}\] 

\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos 2x}\] 

\[\lim_{x \to 0} \frac{\sin \left( 3 + x \right) - \sin \left( 3 - x \right)}{x}\] 

\[\lim_{x \to 0} \frac{\cos 2x - 1}{\cos x - 1}\] 

\[\lim_{x \to 0} \frac{3 \sin^2 x - 2 \sin x^2}{3 x^2}\] 

\[\lim_{x \to 0} \frac{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}{x}\] 

\[\lim_{x \to 0} \frac{1 - \cos 4x}{x^2}\] 

\[\lim_{x \to 0} \frac{x \cos x + \sin x}{x^2 + \tan x}\] 

\[\lim_{x \to 0} \frac{1 - \cos 2x}{3 \tan^2 x}\] 

\[\lim_\theta \to 0 \frac{1 - \cos 4\theta}{1 - \cos 6\theta}\] 

\[\lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}\]

\[\lim_\theta \to 0 \frac{\sin 4\theta}{\tan 3\theta}\] 

Evaluate the following limits: 

\[\lim_{x \to 0} \frac{2\sin x - \sin2x}{x^3}\] 

\[\lim_{x \to 0} \frac{1 - \cos 5x}{1 - \cos 6x}\]

\[\lim_{x \to 0} \frac{cosec x - \cot x}{x}\]

\[\lim_{x \to 0} \frac{\sin 3x + 7x}{4x + \sin 2x}\]