PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

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

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

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

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

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

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

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:

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

\[\lim_{x \to a} \frac{\cos x - \cos a}{x - a}\] 

\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{x - \frac{\pi}{4}}\] 

\[\lim_{x \to \frac{\pi}{2}} \frac{1 - \sin x}{\left( \frac{\pi}{2} - x \right)^2}\]

\[\lim_{x \to \frac{\pi}{3}} \frac{\sqrt{3} - \tan x}{\pi - 3x}\]

\[\lim_{x \to \frac{\pi}{2}} \frac{\cot x}{\frac{\pi}{2} - x}\]

\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{\cos x} - \sqrt{\sin x}}{x - \frac{\pi}{4}}\]