RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Rajasthan Board RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 Textbook Exercise Questions and Answers.

Rajasthan Board RBSE Solutions for Class 11 Maths in Hindi Medium & English Medium are part of RBSE Solutions for Class 11. Students can also read RBSE Class 11 Maths Important Questions for exam preparation. Students can also go through RBSE Class 11 Maths Notes to understand and remember the concepts easily.

RBSE Class 11 Maths Solutions Chapter 13 Limits and Derivatives Ex 13.1

Evaluate the following limits in exercise 1 to 22:

Question 1.
\(\lim _{x \rightarrow 3}\) (x + 3).
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 1

Question 2.
\(\lim _{x \rightarrow \pi}\left(x-\frac{22}{7}\right)\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 2

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 3.
\(\lim _{r \rightarrow 1}\) πr2
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 3

Question 4.
\(\lim _{x \rightarrow 4} \frac{4 x+3}{x-2}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 4

Question 5.
\(\lim _{x \rightarrow(-1)} \frac{x^{10}+x^5+1}{x-1}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 5

Question 6.
\(\lim _{x \rightarrow 0} \frac{(x+1)^5-1}{x}\)
Answer:
Let y = x + 1, when x → 0 then y → 1
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 6
= 5(1)5 - 1
= 5 × 14
= 5

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 7.
\(\lim _{x \rightarrow 2} \frac{3 x^2-x-10}{x^2-4}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 7

Question 8.
\(\lim _{x \rightarrow 3} \frac{x^4-81}{2 x^2-5 x-3}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 8

Question 9.
\(\lim _{x \rightarrow 0} \frac{a x+b}{c x+1}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 9

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 10.
\(\lim _{z \rightarrow 1} \frac{z^{1 / 3}-1}{z^{1 / 6}-1}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 10

Question 11.
\(\lim _{x \rightarrow 1} \frac{a x^2+b x+c}{c x^2+b x+a}\), a + b + c ≠ 0.
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 11

Question 12.
\(\lim _{x \rightarrow(-2)} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 12

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 13.
\(\lim _{x \rightarrow 0} \frac{\sin a x}{b x}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 13

Question 14.
\(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}\), (a, b ≠ 0)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 14

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 15.
\(\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 15

Question 16.
\(\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 16

Question 17.
\(\lim _{x \rightarrow 0} \frac{\cos 2 x-1}{\cos x-1}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 17

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 18.
\(\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 18

Question 19.
\(\lim _{x \rightarrow 0}\) x sec x
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 19

Question 20.
\(\lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x+\sin b x}\); [a, b, a + b ≠ 0]
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 20

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 21.
\(\lim _{x \rightarrow 0}\) (cosec x - cot x)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 21

Question 22.
\(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{\left(x-\frac{\pi}{2}\right)}\)
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 22

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 23.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 23
Answer:
We have,
when x tends to zero, then
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 24
If x > 0 means, x is towards the right hand side of zero.
Thus, it is limit of right side.
Thus, x > 0, then limit of right hand side
= \(\lim _{x \rightarrow 0}\) f(x)
= \(\lim _{x \rightarrow 0}\) [3(x + 1)]
= 3 (0 + 1) [by given function]
= 3 × 1
= 3
When x <0 then L.H.L.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 25

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 24.
Find \(\lim _{x \rightarrow 1}\) f(x), where
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 26
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 27

Question 25.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 28
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 29

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 26.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 30
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 31

Question 27.
Find \(\lim _{x \rightarrow 5}\) f(x) where f(x) = |x| - 5
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 32

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 28.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 33
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 34
then b - a = 4
and a + b = 4
adding both
2b = 8
⇒ b = 4
Again b - a = 4
4 - a = 4
a = 4 - 4 = 0
Thus, a = 0, b = 4

Question 29.
Let a1, a2, ................, an be fixed real numbers and define a function
f(x) = (x - a1) (x - a2) ....... (x - an) what is \(\lim _{x \rightarrow a_1}\)?
For some a ≠ a1, a2, ........ an calculate \(\lim _{x \rightarrow a_1}\).
Answer:
f(x) = (x - a1) (x - a2) ....... (x - an)
where a1, a2, ................ an are real numbers
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 35

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 30.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 36
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 37

Question 31.
If function f(x) satisfies \(\lim _{x \rightarrow 1} \frac{f(x)-2}{x^2-1}\) = π then find for \(\lim _{x \rightarrow 1}\) f(x).
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 38

RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Question 32.
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 39
Answer:
RBSE Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 40

Bhagya
Last Updated on Nov. 17, 2023, 9:43 a.m.
Published Nov. 23, 2022