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

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Evalute the following limits in questions 1 to 22:
1. \(\lim _{x \rightarrow 3}\) (x + 3)
2. \(\lim _{x \rightarrow Ï€}\) (x – \(\frac{22}{7}\))
3. \(\lim _{r \rightarrow 1}\) (Ï€r2)
4. \(\lim _{x \rightarrow 4}\) \(\frac{4x+3}{x-2}\)
5. \(\lim _{x \rightarrow -1}\) \(\frac{x^{10}+x^{5}+1}{x-1}\)
6. \(\lim _{x \rightarrow 0}\) \(\frac{(x+1)^{5}-1}{x}\)
7. \(\lim _{x \rightarrow 2}\) \(\frac{3 x^{2}-x-10}{x^{2}-4}\)
8. \(\lim _{x \rightarrow 3}\) \(\frac{x^{4}-81}{2x^{2}-5x-3}\)
9. \(\lim _{x \rightarrow 0}\) \(\frac{ax+b}{cx+1}\)
10. \(\lim _{z \rightarrow 1}\) \(\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}\)
11. \(\lim _{x \rightarrow 1}\) \(\frac{ax^{2}+bx+c}{cx^{2}+bx+a}\)
12. \(\lim _{x \rightarrow -2}\) \(\frac{1/x+1/2}{x+2}\)
13. \(\lim _{x \rightarrow 0}\) \(\frac{sin ax}{bx}\)
14. \(\lim _{x \rightarrow 0}\) \(\frac{sin ax}{bx}\), ab ≠ 0
15. \(\lim _{x \rightarrow π}\) \(\frac{sin (π-x)}{π(π-x)}\)
16. \(\lim _{x \rightarrow 0}\) \(\frac{cos x}{Ï€-x}\)
17. \(\lim _{x \rightarrow 0}\) \(\frac{cos 2x-1}{cos x-1}\)
18. \(\lim _{x \rightarrow 0}\) \(\frac{ax+xcosx}{bsinx}\)
19. \(\lim _{x \rightarrow 0}\) (x sec x)
20. \(\lim _{x \rightarrow 0}\) \(\frac{sin ax+bx}{ax+sinbx}\), a, b, a + b ≠ 0
21. \(\lim _{x \rightarrow 0}\) (cosec x – cot x)
22. \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 \pi}{x-\frac{\pi}{2}}\)
Solutions to questions 1 to 22:
1. \(\lim _{x \rightarrow 3}\) (x + 3) = 3 + 3 = 6

2. \(\lim _{x \rightarrow Ï€}\) (x – \(\frac{22}{7}\)) = Ï€ – \(\frac{22}{7}\)

3. \(\lim _{r \rightarrow 1}\) (πr2) = π.12 = π.

4. \(\lim _{x \rightarrow 4}\) \(\frac{4x+3}{x-2}\) = \(\frac{4×4+3}{4-2}\) = \(\frac{16+3}{2}\) = \(\frac{19}{2}\).

5.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 1

6.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 2

7.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 3

8.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 4

9. \(\lim _{x \rightarrow 0}\) \(\frac{ax+b}{cx+1}\) = \(\frac{0+b}{0+1}\) = \(\frac{b}{1}\) = b.

10.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 5
= (1)1/6 + 1
= 1 + 1 = 2.

11.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 6

12.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 7

13. \(\lim _{x \rightarrow 0}\) \(\frac{sin ax}{bx}\) = \(\lim _{x \rightarrow 0}\). \(\frac{sin ax}{b.ax}\).a = \(\lim _{x \rightarrow 0}\) \(\frac{sin ax}{ax}\). \(\frac{a}{b}\) ) = \(\frac{a}{b}\).

14.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 8

15. \(\lim _{x \rightarrow π}\) \(\frac{sin (π-x)}{π(π-x)}\).
Put Ï€ – x = θ, As x → Ï€, θ → 0 (zero)

16. \(\lim _{x \rightarrow 0}\) \(\frac{cos x}{Ï€-x}\) = \(\frac{cos 0}{Ï€-0}\) = \(\frac{1}{Ï€}\).

17.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 9

18.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 10

19.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 11

20.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 12

21.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 13

22.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 14

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

Question 23.
Find \(\lim _{x \rightarrow 0}\) f(x) and \(\lim _{x \rightarrow 1}\) f(x), where
f(x) = \(\left\{\begin{array}{l}
2 x+3, x \leq 0 \\
3(x+1), x>0
\end{array}\right.\)
Solution:
To find \(\lim _{x \rightarrow 0^{-}}\) f(x), we have to find \(\lim _{x \rightarrow 0^{-}}\) (2x + 3).
Here, we put values of x near the zero but less than zero, i.e;
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 15

Alternative method:
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 16

Question 24.
Find \(\lim _{x \rightarrow 1}\) f(x), where f(x) = \(\left\{\begin{array}{l}
x^{2}-1, x \leq 1 \\
-x^{2}-1, x>1
\end{array}\right.\)
Solution:
When x ≤ 1, f(x) = x2 – 1:
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 17
∴ \(\lim _{x \rightarrow 1}\) f(x) does not exist.

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

Question 25.
Evalute \(\lim _{x \rightarrow 0}\) f(x), where f(x) = \(\left\{\begin{array}{l}
\frac{|x|}{x}, x \neq 0 \\
0, \quad x=0
\end{array}\right.\).
Solution:
When x < 0, |x| = – x.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 18
i.e; \(\lim _{x \rightarrow 0}\) f(x) does not exist.

Question 26.
Find \(\lim _{x \rightarrow 0}\) f(x), where f(x) = \(\left\{\begin{array}{l}
\frac{x}{|x|}, x \neq 0 \\
0, x=0
\end{array}\right.\).
Solution:
When x < 0, |x| = – x image 19 ∴ \(\lim _{x \rightarrow 0}\) f(x) does not exist.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 19

Question 27.
Find \(\lim _{x \rightarrow 5}\) f(x), where f(x) = |x| – 5.
Solution: When x > 5, put x = 5 + h, where h is small
So, |x| = |5 + h| = 5 + h.
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 20

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

Question 28.
Suppose f(x) = \(\left\{\begin{array}{l}
a+b x, x<1 \\ 4, x=1 \\ b-a x, x>1
\end{array}\right.\) and if \(\lim _{x \rightarrow 1}\) f(x) = f(1), what are possible values of a and b?
Solution:
When x < 1, f(x) = a + bx
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 21
Adding (1) and (2), we get
2b = 8 ⇒ b = 4
and a = 0.
Then, a = 0 and b = 4.

Question 29.
Let a1, a2, ………………. an be fixed real numbers and define a
f(x) = (x – a1)(x – a2) ………………… (x – an).
What is the \(\lim _{x \rightarrow a_{1}}\) f(x)? For some a ≠ a1, a2, ………….., an, compute \(\lim _{x \rightarrow a}\) f(x).
Solution:
(i) Consider the factor x – a1, As x → a1, x – a1 → 0
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 22
When a ≠ a1, a2, a3, …………………
Consider the factor x – a1, As x → a, x – a1 → a – a1.
a – a1 is neither zero nor indeterminate. It is a unique value. So, is the case with other factors and their values, the (a – a2)1 (a – a3), ………….. (a – an)
∴ \(\lim _{x \rightarrow a}\) f(x) = \(\lim _{x \rightarrow a}\) (x – a1)(x – a2) …………. : (x – an)
= (a – a1)(a – a2) …………… (a – an).

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

Question 30.
If f(x) = \(\left\{\begin{array}{l}
|x|+1, x<0 \\ 0, x=0 \\ |x|-1, x>0
\end{array}\right.\), then
Solution:
(i) consider the limit at x = 0
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 23
∴ \(\lim _{x \rightarrow 0}\) f(x) does not exist.

(ii) When x ≠ 0, let x = a (a < 0)
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 24
At all points x < a, \(\lim _{x \rightarrow a}\) f(x) exists. (iii) When a > 0, f(x) = |x| – 1 = x – 1
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 25
Thus, \(\lim _{x \rightarrow a}\) f(x) does not exist and it is equal to a – 1.
When a < 0, \(\lim _{x \rightarrow a}\) f(x) = 1 – a. When a > 0, \(\lim _{x \rightarrow a}\) f(x) = a – 1.

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

Question 31.
If the function f(x) satisfies \(\lim _{x \rightarrow 1}\) \(\frac{f(x)-2}{x^{2}-1}\) = π, evaluate \(\lim _{x \rightarrow 1}\) f(x).
Solution:
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 26

Question 32.
For what integers m and n does both \(\lim _{x \rightarrow 0}\) f(x) and \(\lim _{x \rightarrow 1}\) f(x) exist for the following function?
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 27
Solution:
(i) When x < 0, f(x) = mx2 + n
GSEB Solutions Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1 img 28

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