Limits and Deri...

\(\lim\limits_{x \to \pi} \frac{sin x}{x - \pi} \) is

\(\lim\limits_{x \to 0} \frac{x^2 cos x}{1-cosx}\) is

\(\lim\limits_{x \to 0} \frac{(1 + x)^n - 1}{x}\) is

\(\lim \limits_{x \to 1} \frac{x^m - 1}{x^n - 1}\) is

\(\lim\limits_{\theta \to 0} \frac{1 - cos 4 \theta}{1 - cos 6 \theta} \) is

\(\lim\limits_{x \to 0} \frac{cosec x - cot x}{x}\) is

\( \lim\limits_{x \to 0} \frac{sin x}{\sqrt{x+1} - \sqrt{1-x}}\) is

\(\lim\limits_{x \to \frac{\pi}{4}} \frac{sec^2 x - 2}{tan x - 1}\) is

\(\lim\limits_{x \to 1} \frac{(\sqrt x - 1)(2x -3)}{2x^2 + x -3}\) is

If f(x) = \( \begin{cases} \frac{sin[x]}{[x]}, & \quad [x] \neq 0 \\ 0, & \quad [x] = 0 \end{cases} \), where [.] denotes the greatest integer function, then \(\lim\limits_{x \to 0}\) f(x) is equal to

\(\lim\limits_{x \to 0} \frac{|sin x|}{x} \) is

Let \(f(x) = \begin{cases} x^2 - 1, & \quad 0 < x < 2 \\ 2x + 3, & \quad 2 \leq x < 3 \end{cases}\), the quadratic equation whose roots are \(\lim\limits_{x \to 2^-}\) f(x) and \(\lim\limits_{x \to 2^+} \) f(x) is

\(\lim\limits_{x \to 0} \frac{tan 2x - x}{3x - sin x}\) is

Let f (x) = x – [x]; ∈ R, then f' \(\frac{1}{2}\) is

If \(y = \sqrt x + \frac{1}{\sqrt x}\), then \(\frac{dy}{dx}\) at x = 1 is

If f(x) = \(\frac{x - 4}{2 \sqrt x}\), then f ′(1) is

If \(y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\), then \(\frac{dy}{dx}\) is

If \(y = \frac{sin x + cosx}{sinx - cosx}\), then \(\frac{dy}{dx}\) at x = 0 is

If \(y = \frac{sin(x + 9)}{cos x}\), then \(\frac{dy}{dx} \) at x = 0 is

If f(x) = \(1 + x + \frac{x^2}{2} + \dots + \frac{x^{100}}{100}\), then f ′(1) is equal to