# Chapter – 13: Limits and Derivatives

# Exercise 13.1

**1. Evaluate the Given limit:**

**Solution: –**

Given

Substituting x = 3, we get

= 3 + 3

= 6

**2. Evaluate the Given limit:**

**Solution: –**

Given limit:

Substituting x = π, we get

= (π – 22 / 7)

**3. Evaluate the Given limit:**

**Solution: –**

Given limit**:**

Substituting r = 1, we get

**= **π(1)^{2}

= π

**4. Evaluate the Given limit:**

**Solution: –**

Given limit:

Substituting x = 4, we get

= [4(4) + 3] / (4 – 2)

= (16 + 3) / 2

= 19 / 2

**5. Evaluate the Given limit:**

**Solution: –**

Given limit:

Substituting x = -1, we get

= [(-1)^{10} + (-1)^{5} + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

**6. Evaluate the Given limit:**

**Solution: –**

Given limit:

= [(0 + 1)^{5} – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

**7. Evaluate the Given limit:**

**Solution: –**

**8. Evaluate the Given limit:**

**Solution: –**

**9. Evaluate the Given limit:**

Solution: –

= [a (0) + b] / c (0) + 1

= b / 1

= b

**10. Evaluate the Given limit: **

**Solution: –**

**11. Evaluate the Given limit:**

**Solution: –**

Given limit:

Substituting x = 1

= [a (1)^{2} + b (1) + c] / [c (1)^{2} + b (1) + a]

= (a + b + c) / (a + b + c)

Given

= 1

**12. Evaluate the Given limit:**

**Solution: –**

By substituting x = – 2, we get

**13. Evaluate the Given limit:**

Solution: –

Given

**14. Evaluate the given limit: **

**Solution: –**

**15. Evaluate the given limit: **

**Solution: –**

**16. Evaluate the given limit: **

**Solution: –**

**17. Evaluate the given limit: **

**Solution: –**

**18. Evaluate the given limit: **

**Solution: –**

**19. Evaluate the given limit: **

**Solution: –**

**20. Evaluate the given limit: **

**Solution: –**

**21. Evaluate the given limit: **

**Solution: –**

**22. Evaluate the given limit: **

**Solution: –**

**23. **

**Solution: –**

**24. Find **

**, where**

**Solution: –**

**25. Evaluate **

**, where f(x) = **

**Solution: –**

**26. Find **

**, where f (x) = **

**Solution: –**

**27. Find **

**, where **

**Solution: –**

**28. Suppose **

**and if **

**what are possible values of a and b**

**Solution: –**

**29. Let a**_{1}**, a**_{2,}**………a**_{n}** be fixed real numbers and define a function**

**f (x) = (x – a**_{1}**) (x – a**_{2}**) ……. (x – a**_{n}**).**

**What is **

**For some a ≠ a _{1}, a_{2}, ……. a_{n}, compute **

**Solution: –**

**30. If **

** For what value (s) of a does**

** exists?**

**Solution: –**

**31. If the function f(x) satisfies**

** , evaluate**

**Solution: –**

**32. If **

** For what integers m and n does both **

**and**

** exist?**

**Solution: –**

# Exercise 13.2

**1. Find the derivative of x**^{2}**– 2 at x = 10**

**Solution: –**

Let f (x) = x^{2} – 2

**2. Find the derivative of x at x = 1.**

**Solution: –**

Let f (x) = x

Then,

**3.** **Find the derivative of 99x at x = l00.**

**Solution: –**

Let f (x) = 99x,

From first principle

= 99

**4. Find the derivative of the following functions from first principle**

**(i) x**^{3}** – 27**

**(ii) (x – 1) (x – 2)**

**(iii) 1 / x**^{2}

**(iv) x + 1 / x – 1**

**Solution: –**

(i) Let f (x) = x^{3} – 27

From first principle

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

(iii) Let f (x) = 1 / x^{2}

From first principle, we get

(iv) Let f (x) = x + 1 / x – 1

From first principle, we get

**5. For the function **

** .Prove that f’ (1) =100 f’ (0).**

**Solution: –**

**6. Find the derivative of **

** for some fixed real number a.**

**Solution: –**

**7. For some constants a and b, find the derivative of(i) (x − a) (x − b)**

**(ii) (ax**^{2}** + b)**^{2}

**(iii) x – a / x – b**

**Solution: –**

(i) (x – a) (x – b)

(ii) (ax^{2} + b)^{2}

= 4ax (ax^{2} + b)

(iii) x – a / x – b

**8.** **Find the derivative of **

** for some constant a.**

**Solution: –**

**9. Find the derivative of**

**(i) 2x – 3 / 4**

**(ii) (5x**^{3}** + 3x – 1) (x – 1)**

**(iii) x**^{-3}** (5 + 3x)**

**(iv) x**^{5}** (3 – 6x**^{-9}**)**

**(v) x**^{-4}** (3 – 4x**^{-5}**)**

**(vi) (2 / x + 1) – x**^{2}** / 3x – 1**

**Solution: –**

(i)

(ii)

(iii)

(iv)

(v)

(vi)

**10. Find the derivative of cos x from first principle**

**Solution: –**

Let f (x) = cos (x + h)

Accordingly, f (x + h) = cos (x +h)

**11. Find the derivative of the following functions:**

**(i) sin x cos x**

**(ii) sec x**

**(iii) 5 sec x + 4 cos x**

**(iv) cosec x**

**(v) 3 cot x + 5 cosec x**

**(vi) 5 sin x – 6 cos x + 7**

**(vii) 2 tan x – 7 sec x**

**Solution: –**

(i) sin x cos x

(ii) sec x

(iii) 5 sec x + 4 cos x

(iv) cosec x

(v) 3 cot x + 5 cosec x

(vi)5 sin x – 6 cos x + 7

Let f (x) = 5 sin x – 6 cos x + 7

(vii) 2 tan x – 7 sec x

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