Chapter 13, Limits and Derivatives Video Solutions, NCERT Class 11 - Math

01:16

Problem 1

Evaluate the following limits
$\lim _{x \rightarrow 3} x+3$

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01:05

Problem 2

Evaluate the following limits
$\lim _{x \rightarrow \pi}\left(x-\frac{22}{7}\right)$

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01:16

Problem 3

Evaluate the following limits
$\lim _{r \rightarrow 1} \pi r^{2}$

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01:06

Problem 4

Evaluate the following limits
$\lim _{x \rightarrow 4} \frac{4 x+3}{x-2}$

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01:07

Problem 5

Evaluate the following limits
$\lim _{x \rightarrow-1} \frac{x^{10}+x^{5}+1}{x-1}$

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03:33

Problem 6

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}$

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02:03

Problem 7

Evaluate the following limits
$\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}-4}$

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02:52

Problem 8

Evaluate the following limits
$\lim _{x \rightarrow 3} \frac{x^{4}-81}{2 x^{2}-5 x-3}$

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01:05

Problem 9

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{a x+b}{c x+1}$

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01:37

Problem 10

Evaluate the following limits
$\lim _{z \rightarrow 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$

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01:11

Problem 11

Evaluate the following limits
$\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c \neq 0$

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01:09

Problem 12

Evaluate the following limits
$\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}$

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01:16

Problem 13

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$

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01:56

Problem 14

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}, a, b \neq 0$

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01:18

Problem 15

Evaluate the following limits
$\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$

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01:03

Problem 16

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}$

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01:31

Problem 17

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{\cos 2 x-1}{\cos x-1}$

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01:19

Problem 18

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}$

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01:01

Problem 19

Evaluate the following limits
$\lim _{x \rightarrow 0} x \sec x$

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03:44

Problem 20

Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x+\sin b x} a, b, a+b \neq 0$

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01:16

Problem 21

Evaluate the following limits
$\lim _{x \rightarrow 0}(\operatorname{cosec} x-\cot x)$

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02:11

Problem 22

Evaluate the following limits
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$

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01:54

Problem 23

Find $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow 1} f(x)$, where $f(x)=\left\{\begin{array}{cc}2 x+3, & x \leq 0 \\ 3(x+1), & x>0\end{array}\right.$

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01:27

Problem 24

Find $\lim _{x \rightarrow 1} f(x)$, where $f(x)=\left\{\begin{array}{cc}x^{2}-1, & x \leq 1 \\ -x^{2}-1, & x>1\end{array}\right.$

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01:27

Problem 25

Evaluate $\lim _{x \rightarrow 0} f(x)$, where $f(x)=\left\{\begin{array}{ll}\frac{|x|}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.$

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01:24

Problem 26

Find $\lim _{x \rightarrow 0} f(x)$, where $f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 0, & x=0\end{array}\right.$

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01:39

Problem 27

Find $\lim _{x \rightarrow 5} f(x)$, where $f(x)=|x|-5$

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01:58

Problem 28

Suppose $f(x)=\left\{\begin{array}{ll}a+b x, & x<1 \\ 4, & x=1 \\ b-a x, & x>1\end{array}\right.$
and if $\lim _{x \rightarrow 1} f(x)=f(1)$ what are possible values of $a$ and $b$ ?

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01:33

Let $a_{1}, a_{2}, \ldots, a_{n}$ be fixed real numbers and define a function
$$
f(x)=\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right) .
$$
What is $\lim _{x \rightarrow a_{1}} f(x) ?$ For some $a \neq a_{1}, a_{2}, \ldots, a_{n}$, compute $\lim _{x \rightarrow a} f(x)$.

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03:10

Problem 30

If $f(x)=\left\{\begin{array}{ll}|x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0\end{array}\right.$
For what value (s) of $a$ does $\lim _{x \rightarrow a} f(x)$ exists?

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01:07

Problem 31

If the function $f(x)$ satisfies $\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi$, evaluate $\lim _{x \rightarrow 1} f(x)$.

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03:20

Problem 32

If $f(x)=\left\{\begin{array}{ll}m x^{2}+n, & x<0 \\ n x+m, & 0 \leq x \leq 1 \\ n x^{3}+m, & x>1\end{array}\right.$. For what integers $m$ and $n$ does both $\lim _{x \rightarrow 0} f(x)$
and $\lim _{x \rightarrow 1} f(x)$ exist?

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