Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1

Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1

Limits and Derivatives Exercise 13.1 Solutions

1. Evaluate the Given limit (x + 3)

Solution

(x + 3) = 3 + 3 = 6

2. Evaluate the Given limit : (x - 22/7)

Solution

3. Evaluate the Given limit :

πr².

Solution

πr² = π(1)² = π

4. Evaluate the Given limit :

(4x + 3)/(x - 2)

Solution

5. Evaluate the Given limit :

(x¹⁰ + x⁵ + 1)/(x – 1)

Solution

6. Evaluate the Given limit :

[(x + 1)⁵ - 1]/x

Solution

7. Evaluate the Given limit :

(3x² - x - 10)/(x² - 4)

Solution

At x = 2, the value of the given rational function takes the form 0/0 .

8. Evaluate the Given limit

(x⁴ -81)/(2x² - 5x - 3)

Solution

At x = 2, the value of the given rational function takes the form 0/0 .

9. Evaluate the Given limit :

(ax + b)/(cx + 1)

Solution

10. Evaluate the Given limit :

Solution

At z = 1, the value of the given function takes the form 0/0 .
Put z^1/6 = x so that z →1 as x →1.

11. Evaluate the Given limit

(ax² + bx + c)/(cx² + bx + a) , a + b + c ≠ 0 .

Solution

= 1 [a + b + c ≠ 0 ]

12. Evaluate the Given limit :

Solution

At x = -2, the value of the given function takes the form 0/0 .

13. Evaluate the Given limit

sin ax/bx

Solution

sin ax/bx
At x = 0, the value of the given function takes the form 0/0 .

14. Evaluate the Given limit :

sin ax/sin bx, a,b ≠ 0

Solution

(sin ax/sin bx), a,b ≠ 0
At x = 0, the value of the given function takes the form 0/0.

15. Evaluate the Given limit

[sin(π -x)]/π(π - x)

Solution

16. Evaluate the given limit

cos x/(π - x)

Solution

17. Evaluate the Given limit :

(cos 2x - 1)/(cos x- 1)

Solution

(cos 2x - 1)/(cos x- 1)
At x = 0, the value of the given function takes the form 0/0 .
Now,

18. Evaluate the Given limit

(ax + x cos x)/b sin x

Solution

(ax + x cos x)/b sin x
At x = 0, the value of the given function takes the form 0/0.
Now,

19. Evaluate the Given limit

sec x

Solution

20. Evaluate the Given limit

(sin ax + bx)/(ax + sin bx) a,b, a+b ≠ 0.

Solution

At x = 0, the value of the given function takes the form 0/0 .
Now,

21. Evaluate the Given limit (cosec x - cot x)

Solution

At x = 0, the value of the given function takes the form ∞→∞.
Now,

22. Evaluate the Given limit

Solution

At x = π/2 , the value of the given function takes the form 0/0.
Now, put x - π/2 = y so that x → π/2, y → 0.

23. Find f(x) and

f(x) where

Solution

The given function is

24. Find

f(x) where

Solution

The given function is

25. Evaluate

f(x) where

Solution

It is observed that

f(x) ≠

f(x).
Hence,

f(x) does not exist.

26. Find

f(x) where

Solution

The given function is

It is observed that

f(x) ≠

f(x).
Hence,

f(x) does not exist .

27. Find

f(x) , where f(x) = |x| - 5

Solution

The given functions is f(x) = |x| - 5.

28. Suppose

and if

f(x) = f(1) what are possible values of a and b?

Solution

The given function is

f(x) =

(a + bx) = a + b

f(x) =

(b - ax) = b - a
f(1) = 4
It is given that

f(x) = f(1).
∴

f(x) =

f(x) =

f(x) = f(1)
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.

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

f(x) ? For some a ≠ a₁, a₂, ..., a_n, compute

Solution

The given function is f(x) = (x – a₁)(x – a₂) … (x - a_n).

30. If

For what value (s) of a does

f(x) exists ?

Solution

The given function is

When a = 0,

31. If the function f(x) satisfies

[f(x) - 2]/[x² -1] = π , evaluate

f(x).

Solution

32. If

For what integers m and n does

f(x) and

f(x) exist ?

Solution

The given function is