NCERT Solutions
Class 11 - Mathematics

Chapter 12: LIMITS AND DERIVATIVES

Evaluate \(\lim_{x\to 3} (x+3)\).

Evaluate \(\lim_{x\to \pi} (x - \tfrac{22}{7})\).

Evaluate \(\lim_{r\to 1} \pi r^{2}\).

Evaluate \(\lim_{x\to 4} \dfrac{4x+3}{x-2}\).

Evaluate \(\lim_{x\to 1} \dfrac{x^{10}+x^{5}+1}{x-1}\).

Evaluate \(\lim_{x\to 0} \dfrac{(x+1)^{5}-1}{x}\).

Evaluate \(\lim_{x\to 2} \dfrac{3x^{2}-x-10}{x^{2}-4}\).

Evaluate \(\lim_{x\to 3} \dfrac{x^{4}-81}{2x^{2}-5x-3}\).

Evaluate \(\lim_{x\to 0} \dfrac{ax+b}{cx+1}\).

Evaluate \(\lim_{z\to 1} \dfrac{z^{1/3}-1}{z^{1/6}-1}\).

Evaluate \(\lim_{x\to 1} \dfrac{ax^{2}+bx+c}{cx^{2}+bx+a}\), where \(a+b+c\neq 0\).

Evaluate \(\lim_{x\to -2} \dfrac{\tfrac{1}{x}+\tfrac{1}{2}}{x+2}\).

Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{bx}\).

Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{\sin bx}\), where \(a,b\neq 0\).

Evaluate \(\lim_{x\to \pi} \dfrac{\sin(\pi - x)}{\pi(\pi - x)}\).

Evaluate \(\lim_{x\to 0} \dfrac{\cos x}{\pi - x}\).

Evaluate \(\lim_{x\to 0} \dfrac{\cos 2x - 1}{\cos x - 1}\).

Evaluate \(\lim_{x\to 0} \dfrac{ax + x\cos x}{b\sin x}\).

Evaluate \(\lim_{x\to 0} x\sec x\).

Evaluate \(\lim_{x\to 0} \dfrac{\sin ax + bx}{ax + \sin bx}\), where \(a, b, a+b \neq 0\).

Evaluate \(\lim_{x\to 0} (\csc x - \cot x)\).

Evaluate \(\lim_{x\to \pi/2} \dfrac{\tan 2x}{x - \pi/2}\).

Find \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\), where

\(f(x) = \begin{cases} 2x+3, & x \le 0 \\ 3(x+1), & x>0 \end{cases}\).

Find \(\lim_{x\to 1} f(x)\), where

\(f(x) = \begin{cases} x^{2}-1, & x \le 1 \\ -x^{2}-1, & x>1 \end{cases}\).

Evaluate \(\lim_{x\to 0} f(x)\), where

\(f(x) = \begin{cases} \dfrac{|x|}{x}, & x \ne 0 \\ 0, & x=0 \end{cases}\).

Find \(\lim_{x\to 0} f(x)\), where

\(f(x) = \begin{cases} \dfrac{x}{|x|}, & x \ne 0 \\ 0, & x=0 \end{cases}\).

Find \(\lim_{x\to 5} f(x)\), where \(f(x) = |x| - 5\).

Suppose

\(f(x) = \begin{cases} a+bx, & x<1 \\ 4, & x=1 \\ b-ax, & x>1 \end{cases}\)

and if \(\lim_{x\to 1} f(x) = f(1)\), what are possible values of \(a\) and \(b\)?

Let \(a_{1}, a_{2}, \ldots, a_{n}\) be fixed real numbers and define

\(f(x) = (x-a_{1})(x-a_{2})\cdots(x-a_{n})\).

What is \(\lim_{x\to a_{1}} f(x)\)? For some \(a \ne a_{1}, a_{2}, \ldots, a_{n}\), compute \(\lim_{x\to a} f(x)\).

If

\(f(x) = \begin{cases} |x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0 \end{cases}\),

for what value(s) of \(a\) does \(\lim_{x\to a} f(x)\) exist?

If the function \(f(x)\) satisfies

\(\displaystyle \lim_{x\to 1} \dfrac{f(x) - 2}{x^{2} - 1} = \pi\),

evaluate \(\lim_{x\to 1} f(x)\).

If

\(f(x) = \begin{cases} mx^{2}+n, & x<0 \\ nx+m, & 0 \le x \le 1 \\ nx^{3}+m, & x>1 \end{cases}\).

For what integers \(m\) and \(n\) does both \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\) exist?

Find the derivative of \(x^{2}-2\) at \(x=10\).

Find the derivative of \(x\) at \(x=1\).

Find the derivative of \(99x\) at \(x=100\).

Find the derivative of the following functions from first principle:

(i) \(x^{3} - 27\)

(ii) \((x-1)(x-2)\)

(iii) \(\dfrac{1}{x^{2}}\)

(iv) \(\dfrac{x+1}{x-1}\)

For the function

\(f(x)=\dfrac{x^{100}}{100}+\dfrac{x^{99}}{99}+\cdots+\dfrac{x^{2}}{2}+x+1\),

prove that \(f'(1)=100f'(0)\).

Find the derivative of the expression:

\(x^{n}+ax^{n-1}+a^{2}x^{n-2}+\cdots +a^{n-1}x+a^{n}\)

For some constants \(a\) and \(b\), find the derivative of the following:

(i) \((x-a)(x-b)\)

(ii) \((ax^{2}+b)^{2}\)

(iii) \(\dfrac{x-a}{x-b}\)

Find the derivative of

\(\dfrac{x^{n}-a^{n}}{x-a}\)

for constant \(a\).

Find the derivative of the following:

(i) \(2x - \dfrac{3}{4}\)

(ii) \((5x^{3}+3x-1)(x-1)\)

(iii) \(x^{-3}(5+3x)\)

(iv) \(x^{5}(3-6x^{-9})\)

(v) \(x^{-4}(3-4x^{-5})\)

(vi) \(\dfrac{2}{x+1} - \dfrac{x^{2}}{3x-1}\)

Find the derivative of \(\cos x\) from first principle.

Find the derivative of the following functions:

(i) \(\sin x \cos x\)

(ii) \(\sec x\)

(iii) \(5\sec x + 4\cos x\)

(iv) \(\csc x\)

(v) \(3\cot x + 5\cosec x\)

(vi) \(5\sin x - 6\cos x + 7\)

(vii) \(2\tan x - 7\sec x\)

Find the derivative of the following functions from first principle:

(i) \(-x\)

(ii) \((-x)^{-1}\)

(iii) \(\sin (x+1)\)

(iv) \(\cos \left(x-\tfrac{\pi}{8}\right)\)

Find the derivative of \(x+a\).

Find the derivative of \((px+q)\left(\dfrac{r}{x}+s\right)\), where \(p,q,r,s\) are fixed non-zero constants.

Find the derivative of \((ax+b)(cx+d)^{2}\).

Find the derivative of \(\dfrac{ax+b}{cx+d}\).

Find the derivative of \(\dfrac{1+\tfrac{1}{x}}{1-\tfrac{1}{x}}\).

Find the derivative of \(\dfrac{1}{ax^{2}+bx+c}\).

Find the derivative of \(\dfrac{ax+b}{px^{2}+qx+r}\).

Find the derivative of \(\dfrac{px^{2}+qx+r}{ax+b}\).

Find the derivative of \(\dfrac{a}{x^{4}}-\dfrac{b}{x^{2}}+\cos x\).

Find the derivative of \(4\sqrt{x}-2\).

Find the derivative of \((ax+b)^{n}\).

Find the derivative of \((ax+b)^{n}(cx+d)^{m}\).

Find the derivative of \(\sin(x+a)\).

Find the derivative of \(\csc x\cot x\).

Find the derivative of \(\dfrac{\cos x}{1+\sin x}\).

Find the derivative of \(\dfrac{\sin x+\cos x}{\sin x-\cos x}\).

Find the derivative of \(\dfrac{\sec x-1}{\sec x+1}\).

Find the derivative of \(\sin^{n}x\).

Find the derivative of \(\dfrac{a+b\sin x}{c+d\cos x}\).

Find the derivative of \(\dfrac{\sin(x+a)}{\cos x}\).

Find the derivative of \(x^{4}(5\sin x-3\cos x)\).

Find the derivative of \((x^{2}+1)\cos x\).

Find the derivative of \((ax^{2}+\sin x)(p+q\cos x)\).

Find the derivative of \((x+\cos x)(x-\tan x)\).

Find the derivative of \(\dfrac{4x+5\sin x}{3x+7\cos x}\).

Find the derivative of \(\dfrac{x^{2}\cos\left(\tfrac{\pi}{4}\right)}{\sin x}\).

Find the derivative of \(\dfrac{x}{1+\tan x}\).

Find the derivative of \((x+\sec x)(x-\tan x)\).

Find the derivative of \(\dfrac{x}{\sin^{n}x}\).