PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

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

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

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

\[\lim_{x \to 0} \frac{\log \left( 2 + x \right) + \log 0 . 5}{x}\]

The standard deviation of the data:

is

\[\lim_{x \to 0} \frac{x\left( 2^x - 1 \right)}{1 - \cos x}\] 

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{\log \left( 1 + x \right)}\] 

\[\lim_{x \to 0} \frac{\log \left| 1 + x^3 \right|}{\sin^3 x}\] 

`\lim_{x \to \pi/2} \frac{a^\cot x - a^\cos x}{\cot x - \cos x}`

\[\lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}}\]

\[\lim_{x \to 5} \frac{e^x - e^5}{x - 5}\]

\[\lim_{x \to 0} \frac{e^{x + 2} - e^2}{x}\] 

`\lim_{x \to \pi/2} \frac{e^\cos x - 1}{\cos x}`

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

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

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

`\lim_{x \to 0} \frac{e^\tan x - 1}{\tan x}`

\[\lim_{x \to 0} \frac{e^{bx} - e^{ax}}{x} \text{ where } 0 < a < b\] 

`\lim_{x \to 0} \frac{e^\tan x - 1}{x}`

`\lim_{x \to 0} \frac{e^x - e^\sin x}{x - \sin x}`