PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

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

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

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

Evaluate the following limits: 

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

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

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

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

\[\lim_{x \to 0} \frac{5x + 4 \sin 3x}{4 \sin 2x + 7x}\]

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

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

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

\[\lim_{x \to 0} \left( cosec x - \cot x \right)\]

Evaluate the following limit: 

\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]

Evaluate the following limits: 

\[\lim_{x \to 0} \frac{\cos ax - \cos bx}{\cos cx - 1}\] 

Evaluate the following limit: 

\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\] 

If  \[\lim_{x \to 0} kx  cosec x = \lim_{x \to 0} x  cosec kx,\] 

\[\lim_{x \to \pi} \frac{\sin x}{\pi - x}\]

\[\lim_{x \to \frac{\pi}{2}} \frac{\sin 2x}{\cos x}\] 

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

Evaluate the following limit:

\[\lim_{x \to \frac{\pi}{3}} \frac{\sqrt{1 - \cos6x}}{\sqrt{2}\left( \frac{\pi}{3} - x \right)}\]