Practice Questions 4-Limits and Derivatives-Class XI

\( \lim_{x \to 0} \dfrac{\tan 2x-x}{3x-\sin x}\) is equal to_______

Evaluate: \( \lim_{x \to 1} \dfrac{(\sqrt{x}-1)(2x-3)}{2x^2+x-3}\)

If \( f(x)=\begin{Bmatrix}
\dfrac{\sin [x]}{[x]}, &[x]\neq 0 \\
0, & [x]=0
\end{Bmatrix}\), then \(\lim_{x \to 0} f(x)\) is equal to______

If \(f(x) =\begin{Bmatrix}
x^2-1, & 0< x <2 \\
2x+3, & 2 \leq x <3
\end{Bmatrix} \), then the quadratic equation whose roots are \(\lim_{x \to 2^-} f(x) \) and \( \lim_{x \to 2^+} f(x)\) is_____

If \( f(x)=x-[x]\), then \(f’\left( \dfrac{1}{2} \right) \) is_______

If \( y=\sqrt{x}+\dfrac{1}{\sqrt{x}}\), then \( \dfrac{dy}{dx}\Big |_{x=1}\)  is equal to_____

If \( f(x)=\dfrac{x-4}{2\sqrt{x}}\), then \( f’ (1)\) is equal to______

Solve: \( \lim_{x \to 3^+} \dfrac{x}{[x]}\)

If \( y=\dfrac{\sin (x+9)}{\cos x}\)  then \( \dfrac{dy}{dx}\Big |_{x=0}\)  is equal to_____

\( \lim_{x \to 0} \left( \sin mx \cdot \cot \dfrac{x}{\sqrt{3}} \right)=2\), then \( m\) equals______

Solve: \( \lim_{x \to \frac{\pi}{2}} \dfrac{\cot x-\cos x}{(\pi-2x)^3} \)

Solve: \( \lim_{x \to 0} \dfrac{\sin (\pi \cos^2 x)}{x^2}\)

Evaluate: \( \lim_{x \to 0} \dfrac{(1-\cos 2x)(3+\cos x)}{x \tan 4x}\)

Find \(\lim_{h \to 0} \dfrac{f(2h+2+h^2)-f(2)}{f(h-h^2+1)-f(1)} \), given \( f'(2) =6\) and \(f'(1)=4 \)

If \(\lim_{x \to 0} \dfrac{\left\{ (a-n)nx-\tan x \right\}\sin x}{x^2}=0 \), where \( n \)  is a non-zero real number, then \( a \)  is equal to_____

The integer \( n \) for which \(\lim_{x \to 0} \dfrac{(\cos x-1)(\cos x-e^x)}{x^n} \)  is a finite non-zero number, is______

Evaluate: \( \lim_{x \to 0} \dfrac{x\tan 2x-2x\tan x}{(1-\cos 2x)^2}\)

The value of \( \lim_{x \to 0} \dfrac{\sqrt{\dfrac{1}{2}(1-\cos^2x)}}{x}\) is______

\( \lim_{n \to \infty} \left( \dfrac{1}{1-n^2}+\dfrac{2}{1-n^2}+…+\dfrac{n}{1-n^2} \right)\) is equal to_____

If \(f(a)=2, ~~f'(a)=1, ~~~g(a)=-1,~ g'(a)=2 \) then the value of \(\lim_{x \to a} \dfrac{g(x) f(a)-g(a)f(x)}{x-a} \) is______

If \( G(x) =-\sqrt{25-x^2}\), then \(\lim_{x \to 1} \dfrac{G(x)-G(1)}{x-1} \) has the value_____

Let \(L=\lim_{x \to 0} \dfrac{a-\sqrt{a^2-x^2}-\dfrac{x^2}{4}}{x^4}, a > 0 \). If \( L \) is finite, then the value of \( L \) is______

Evaluate: \(\lim_{h \to 0} \dfrac{\log (1+2h)-2\log (1+h)}{h^2} \)

If \(f(x)=\begin{Bmatrix}
\sin x, &x \neq n \pi,n=0,\pm1,\pm 2… \\
2, & \text {otherwise}
\end{Bmatrix} \) and \( g(x)=\begin{Bmatrix}
x^2+1, &x \neq0 ,2 \\
4,& x=0\\
5,&x=2
\end{Bmatrix}\), then \(\lim_{x \to 0} g [f(x)]\) is_____

Evaluate: \( \lim_{x \to 1} (1-x)\tan \dfrac{\pi x}{2}\)