NCERT solutions for Mathematics [English] Class 11 chapter 12 - Limits and Derivatives [Latest edition]

Evaluate the following limit.

${\displaystyle \underset{x\to 3}{lim}x+3}$

Evaluate the following limit.

${\displaystyle \underset{x\to \pi }{lim}(x-\frac{22}{7})}$

Evaluate the following limit:

${\displaystyle \underset{r\to l}{lim}\pi r^2}$

Evaluate the following limit.

${\displaystyle \underset{x\to 4}{lim}\frac{4x+3}{x-2}}$

Evaluate the following limit.

${\displaystyle \underset{x\to -1}{lim}\frac{x^{10}+x^5+1}{x-1}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{{(x+1)}^5-1}{x}}$

Evaluate the following limit.

${\displaystyle \underset{x->2}{lim}\frac{3x^2-x-10}{x^2-4}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 3}{lim}\frac{x^4-81}{2x^2-5x-3}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{ax+b}{cx+1}}$

Evaluate the following limit.

${\displaystyle \underset{z\to 1}{lim}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 1}{lim}\frac{a{x}^2+bx +c}{c{x}^2+bx+a},a+b+c\ne 0}$

Evaluate the following limit.

${\displaystyle \underset{x\to -2}{lim}\frac{\frac{1}{x}+\frac{1}{2}}{x+2}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{\sin ax}{bx}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{\sin ax}{\sin bx},a,b\ne 0}$

Evaluate the following limit.

${\displaystyle \underset{x\to \pi }{lim}\frac{\sin (\pi -x)}{\pi (\pi -x)}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{\cos x}{\pi -x}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{\cos 2x-1}{\cos x-1}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{ax+ x\cos x}{b\sin x}}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}x\sec x}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}a,b,a+b\ne 0}$

Evaluate the following limit.

${\displaystyle \underset{x\to 0}{lim}(\cos ecx- \cot x)}$

Evaluate the following limit.

${\displaystyle \underset{x\to \frac{\pi }{2}}{lim}\frac{\tan 2x}{x-\frac{\pi }{2}}}$

Find ${\displaystyle \underset{x\to 0}{lim}}$ f(x) and ${\displaystyle \underset{x\to 1}{lim}}$ f(x) where f(x) = ${\displaystyle \{\begin{array}{ll}2x+3& x\le 0\\ 3(x+1)& x>0\end{array}}$

Find ${\displaystyle \underset{x\to 1}{lim}}$ f(x), where ${\displaystyle f\left(x\right)=\{\begin{array}{ll}{x}^2-1& x\le 1\\ -x^2-1& x>1\end{array}}$

Evaluate ${\displaystyle \underset{x\to 0}{lim}f\left(x\right)}$ where ${\displaystyle f\left(x\right)=\{\begin{array}{ll}\frac{\left|x\right|}{x}& x\ne 0\\ 0& x=0\end{array}}$

Find ${\displaystyle \underset{x\to 0}{lim}}$ f(x), where ${\displaystyle f\left(x\right)=\{\begin{array}{ll}\frac{x}{\left|x\right|}& x\ne 0\\ 0& x=0\end{array}}$

Find ${\displaystyle \underset{x\to 5}{lim}f\left(x\right)}$, where f(x)  = |x| - 5

Suppose f(x)  = ${\displaystyle \{\begin{array}{ll}a+bx& x<1\\ 4& x=1\\ b-ax& x>1\end{array}}$  and if ${\displaystyle \underset{x\to 1}{lim}f\left(x\right)=f\left(1\right)}$ what are possible values of a and b?

Let a₁, a₂,..., a_n be fixed real numbers and define a function f ( x) = ( x − a₁ ) ( x − a₂ )...( x − a_n ).

What is ${\displaystyle \underset{x\to a_1}{lim}f\left(x\right)}$ ? For some a ≠ a₁, a₂, ..., a_n, compute ${\displaystyle \underset{x\to a}{lim}f\left(x\right)}$

If f(x) = ${\displaystyle \{\begin{array}{ll}\left|x\right|+ 1& x<0\\ 0& x=0\\ \left|x\right|-1& x>0\end{array}}$

For what value (s) of a does ${\displaystyle \underset{x\to a}{lim}}$  f(x) exists?

If the function f(x) satisfies ${\displaystyle \underset{x\to 1}{lim}\frac{f\left(x\right)-2}{x^2-1}=\pi }$, evaluate ${\displaystyle \underset{x\to 1}{lim}f\left(x\right)}$.

if ${\displaystyle f\left(x\right)=\{\begin{array}{ll}m{x}^2+n& x<0\\ nx+m& 0\le x\le 1\\ n{x}^3+m& x>1\end{array}}$

For what integers m and n does ${\displaystyle \underset{x\to 0}{lim}f\left(x\right)}$ and ${\displaystyle \underset{x\to 1}{lim}f\left(x\right)}$ exist?

Find the derivative of x² – 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 function from first principle.

x³ – 27

Find the derivative of the following function from first principle.

(x – 1) (x – 2)

Find the derivative of the following function from first principle.

${\displaystyle \frac{1}{x^2}}$

Find the derivative of the following function from first principle.

${\displaystyle \frac{x+1}{x-1}}$

For the function

f(x) = ${\displaystyle \frac{x^{100}}{100}+\frac{x^{99}}{99}+...+\frac{x^2}{2}+x+1}$

Prove that f'(1) = 100 f'(0)

Find the derivative of ${\displaystyle x^n +a{x}^{n-1}+a^2{x}^{n-2}+...+a^{n-1}x+a^n}$ for some fixed real number a.

For some constants a and b, find the derivative of (x – a) (x – b).

For some constants a and b, find the derivative of (ax² + b)².

For some constants a and b, find the derivative of ${\displaystyle \frac{x-a}{x-b}}$.

Find the derivative of ${\displaystyle \frac{x^n-a^n}{x-a}}$ for some constant a.

Find the derivative of ${\displaystyle 2x-\frac{3}{4}}$

Find the derivative of (5x³ + 3x – 1) (x – 1).

Find the derivative of x^–3 (5 + 3x).

Find the derivative of x⁵ (3 – 6x^–9).

Find the derivative of x^–4 (3 – 4x^–5).

Find the derivative of ${\displaystyle \frac{2}{x+1}-\frac{x^2}{3x-1}}$.

Find the derivative of cos x from first principle.

Find the derivative of the following function.

sin x cos x 

Find the derivative of the following function:

sec x

Find the derivative of the following function:

5 sec x + 4 cos x

Find the derivative of the following function:

cosec x

Find the derivative of the following function:

3cot x + 5cosec x

Find the derivative of the following function:

5sin x – 6cos x + 7

Find the derivative of the following function:

2tan x – 7sec x

Find the derivative of the following function from first principle:

−x

Find the derivative of the following function from first principle:

(–x)^–1

Find the derivative of the following function from first principle: 

sin (x + 1)

Find the derivative of the following function from first principle: 

${\displaystyle \cos (x-\frac{\pi }{8})}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(x + a)

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle (px+q)(\frac{r}{s}+s)}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(ax + b) (cx + d)²

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{ax+b}{cx+d}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{1+\frac{1}{x}}{1-\frac{1}{x}}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{1}{a{x}^2+bx+c}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{ax+b}{p{x}^2+qx+r}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{p{x}^2+qx+r}{ax+b}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{a}{x^4}=\frac{b}{x^2}+\cos x}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle 4\sqrt{x}-2}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(ax + b)ⁿ

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(ax + b)ⁿ (cx + d)^m

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

sin (x + a)

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

cosec x cot x

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{\cos x}{1+\sin x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{\sin x+\cos x}{\sin x-\cos x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{\sec x-1}{\sec x+1}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

sinⁿ x

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{a+b\sin x}{c+d\cos x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{\sin (x+a)}{\cos x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

x⁴ (5 sin x – 3 cos x)

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(x² + 1) cos x

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(ax² + sin x) (p + q cos x)

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle (x+\cos x)(x-\tan x)}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{4x+5\sin x}{3x+7\cos x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{x^2\cos \left(\frac{\pi }{4}\right)}{\sin x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{x}{1+\tan x}}$

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

(x + sec x) (x – tan x)

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

${\displaystyle \frac{x}{{\sin }^{n}x}}$