Limits and Derivatives Class 11 Notes

• The limit exists only when the left-hand and right-hand limits are equal.
• Graphically, the limit is the y-value the curve approaches, regardless of whether the point is plotted.
• Limits form the foundation for both continuity and differentiation.

• Sum rule:
• $\lim (f+g)=\lim f+\lim g$
• .
• Difference rule:
• $\lim (f-g)=\lim f-\lim g$
• .
• Product rule:
• $\lim (f\cdot g)=\lim f\cdot \lim g$
• .
• Quotient rule:
• $\lim \frac{f}{g}=\frac{\lim f}{\lim g}$
• provided
• $\lim g\ne 0$
• .
• Scalar multiple:
• $\lim (kf)=k\lim f$
• .

• Trigonometric limit:
• ${\lim }_{x\to 0}\frac{\sin x}{x}=1$
• .
• Cosine limit:
• ${\lim }_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}$
• .
• Power limit:
• ${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$
• for any positive integer n.
• Exponential limit:
• ${\lim }_{x\to 0}\frac{e^x-1}{x}=1$
• .

• A function is continuous on an interval if it is continuous at every point in that interval.
• Polynomials, rational functions, trigonometric functions and exponentials are continuous in their domains.
• A jump, hole or vertical asymptote makes the function discontinuous at that point.

• If the limit does not exist, f is not differentiable at that point.
• Differentiability implies continuity, but continuity does not guarantee differentiability (e.g. |x| at x = 0).
• The derivative is also written as dy/dx or df/dx.

• Sum rule:
• $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$
• .
• Difference rule:
• $(f-g{)}^{\prime }=f^{\prime }-g^{\prime }$
• .
• Constant multiple:
• $(cf{)}^{\prime }=c{f}^{\prime }$
• .
• Product rule:
• $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$
• .
• Quotient rule:
• ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$
• , provided v ≠ 0.

• Constant:
• $\frac{d}{dx}(c)=0$
• .
• Power rule:
• $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• for any real n.
• Trigonometric:
• $\frac{d}{dx}(\sin x)=\cos x$
• ,
• $\frac{d}{dx}(\cos x)=-\sin x$
• ,
• $\frac{d}{dx}(\tan x)={\sec }^{2}x$
• .
• Exponential and logarithmic:
• $\frac{d}{dx}(e^x)=e^x$
• ,
• $\frac{d}{dx}(\ln x)=\frac{1}{x}$
• .

Example: Find ${\lim }_{x\to 2}\frac{x^3-8}{x-2}$.

Solution: Factor the numerator as $(x-2)(x^2+2x+4)$. Cancel (x − 2) to get $x^2+2x+4$. Substituting x = 2 gives 4 + 4 + 4 = 12.

Example: Differentiate $f(x)=x^2\sin x$ using the product rule.

Solution: Let u = x² and v = sin x. Then u′ = 2x and v′ = cos x. By the product rule, f′(x) = 2x sin x + x² cos x.

Exercises (NCERT Q1–Q43)

• 1) lim(x→3) (x + 3)
• 2) lim(x→π) (x − 22/7)
• 3) lim(r→1) πr²
• 4) lim(x→4) (4x + 3)/(x − 2)
• 5) lim(x→−1) (x¹⁰ + x⁵ + 1)/(x − 1)
• 6) lim(x→0) ((x + 1)⁵ − 1)/x
• 7) lim(x→2) (3x² − x − 10)/(x² − 4)
• 8) lim(x→3) (x⁴ − 81)/(2x² − 5x − 3)
• 9) lim(x→0) (ax + b)/(cx + 1)
• 10) lim(z→1) (z¹ᐟ³ − 1)/(z¹ᐟ⁶ − 1)
• 11) lim(x→1) (ax² + bx + c)/(cx² + bx + a), where a + b + c ≠ 0
• 12) lim(x→−2) (1/x + 1/2)/(x + 2)
• 13) lim(x→0) sin(ax)/bx
• 14) lim(x→0) sin(ax)/sin(bx), where a, b ≠ 0
• 15) lim(x→π) sin(π − x)/(π(π − x))
• 16) lim(x→0) cos x/(π − x)
• 17) lim(x→0) (cos 2x − 1)/(cos x − 1)
• 18) lim(x→0) (ax + x cos x)/(b sin x)
• 19) lim(x→0) x sec x
• 20) lim(x→0) (sin ax + bx)/(ax + sin bx), where a, b, a + b ≠ 0
• 21) lim(x→0) (cosec x − cot x)
• 22) lim(x→π/2) tan 2x/(x − π/2)
• 23) Find lim(x→0) f(x) and lim(x→1) f(x), where f(x) = {2x + 3, x ≤ 0; 3(x + 1), x > 0}.
• 24) Find lim(x→1) f(x), where f(x) = {x² − 1, x ≤ 1; −x² − 1, x > 1}.
• 25) Evaluate lim(x→0) f(x), where f(x) = {|x|/x, x ≠ 0; 0, x = 0}.
• 26) Evaluate lim(x→0) f(x), where f(x) = {x/|x|, x ≠ 0; 0, x = 0}.
• 27) Find lim(x→5) f(x), where f(x) = |x| − 5.
• 28) Suppose f(x) = {a + bx, x < 1; 4, x = 1; b − ax, x > 1} and if lim(x→1) f(x) = f(1), what are the possible values of a and b?
• 29) Let a₁, a₂, …, aₙ be fixed real numbers and define f(x) = (x − a₁)(x − a₂)…(x − aₙ). What is lim(x→a₁) f(x)? For some a ≠ a₁, a₂, …, aₙ, compute lim(x→a) f(x).
• 30) If f(x) = {|x| + 1, x < 0; 0, x = 0; |x| − 1, x > 0}, for what value(s) of a does lim(x→a) f(x) exist?
• 31) If the function f(x) satisfies lim(x→1) (f(x) − 2)/(x² − 1) = π, evaluate lim(x→1) f(x).
• 32) If f(x) = {mx² + n, x < 0; nx + m, 0 ≤ x ≤ 1; nx³ + m, x > 1}, for what integers m and n does both lim(x→0) f(x) and lim(x→1) f(x) exist?
• 33) Find the derivative of x² − 2 at x = 10.
• 34) Find the derivative of x at x = 1.
• 35) Find the derivative of 99x at x = 100.
• 36) Find the derivative of the following functions from first principle: (i) x³ − 27, (ii) (x − 1)(x − 2), (iii) 1/x², (iv) (x + 1)/(x − 1).
• 37) For the function f(x) = x¹⁰⁰/100 + x⁹⁹/99 + … + x²/2 + x + 1, prove that f′(1) = 100 f′(0).
• 38) Find the derivative of xⁿ + axⁿ⁻¹ + a²xⁿ⁻² + … + aⁿ⁻¹x + aⁿ for some fixed real number a.
• 39) For some constants a and b, find the derivative of: (i) (x − a)(x − b), (ii) (ax² + b)², (iii) (x − a)/(x − b).
• 40) Find the derivative of (xⁿ − aⁿ)/(x − a) for some constant a.
• 41) Find the derivative of: (i) 2x − 3/4, (ii) (5x³ + 3x − 1)(x − 1), (iii) x⁻³(5 + 3x), (iv) x⁵(3 − 6x⁻⁹), (v) x⁻⁴(3 − 4x⁻⁵), (vi) 2/(x + 1) − x²/(3x − 1).
• 42) Find the derivative of cos x from first principle.
• 43) 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.

Answers at a glance

1. 1
   Q1 → 6.
2. 2
   Q2 → π − 22/7.
3. 3
   Q3 → π.
4. 4
   Q4 → 19/2.
5. 5
   Q5 → −1/2.
6. 6
   Q6 → 5.
7. 7
   Q7 → 11/4.
8. 8
   Q8 → 108/7.
9. 9
   Q9 → b.
10. 10
    Q10 → 2.
11. 11
    Q11 → 1.
12. 12
    Q12 → −1/4.
13. 13
    Q13 → a/b.
14. 14
    Q14 → a/b.
15. 15
    Q15 → 1/π.
16. 16
    Q16 → 1/π.
17. 17
    Q17 → 4.
18. 18
    Q18 → (a + 1)/b.
19. 19
    Q19 → 0.
20. 20
    Q20 → 1.
21. 21
    Q21 → 0.
22. 22
    Q22 → 2.
23. 23
    Q23 → lim(x→0) = 3, lim(x→1) = 6.
24. 24
    Q24 → Limit does not exist (LHL 0, RHL −2).
25. 25
    Q25 → Does not exist (LHL −1, RHL 1).
26. 26
    Q26 → Does not exist (LHL −1, RHL 1).
27. 27
    Q27 → 0.
28. 28
    Q28 → a = 0, b = 4.
29. 29
    Q29 → lim(x→a₁) = 0; lim(x→a) = (a − a₁)(a − a₂)…(a − aₙ).
30. 30
    Q30 → The limit exists for every a ≠ 0.
31. 31
    Q31 → 2.
32. 32
    Q32 → m = n (any integer pair).
33. 33
    Q33 → 20.
34. 34
    Q34 → 1.
35. 35
    Q35 → 99.
36. 36
    Q36 → (i) 3x², (ii) 2x − 3, (iii) −2/x³, (iv) −2/(x − 1)².
37. 37
    Q37 → f′(1) = 100, f′(0) = 1 → f′(1) = 100 f′(0). Proved.
38. 38
    Q38 → nxⁿ⁻¹ + (n−1)axⁿ⁻² + (n−2)a²xⁿ⁻³ + … + aⁿ⁻¹.
39. 39
    Q39 → (i) 2x − a − b, (ii) 4a²x³ + 4abx, (iii) (a − b)/(x − b)².
40. 40
    Q40 → [(n − 1)xⁿ − naxⁿ⁻¹ + aⁿ]/(x − a)².
41. 41
    Q41 → (i) 2, (ii) 20x³ − 15x² + 6x − 4, (iii) −15/x⁴ − 6/x³, (iv) 15x⁴ + 24/x⁵, (v) −12/x⁵ + 36/x¹⁰, (vi) −2/(x+1)² − (3x²−2x)/(3x−1)².
42. 42
    Q42 → −sin x.
43. 43
    Q43 → (i) cos 2x, (ii) sec x tan x, (iii) 5 sec x tan x − 4 sin x, (iv) −cosec x cot x, (v) −3cosec²x − 5 cosec x cot x, (vi) 5 cos x + 6 sin x, (vii) 2sec²x − 7 sec x tan x.

MCQ Questions

1 Mark Questions

1. 1
   State lim(x→0)(1 − cos x)/x. (Ans: 0)
2. 2
   Write d/dx (sin x). (Ans: cos x)
3. 3
   Find lim(x→1)(x² − 1)/(x − 1). (Ans: 2)
4. 4
   Write the derivative of xⁿ using the power rule. (Ans: n xⁿ⁻¹)

2 Mark Questions

1. 1
   Evaluate lim(x→0) tan x / x. (Ans: 1)
2. 2
   Find d/dx (3x² − 2x + 1). (Ans: 6x − 2)
3. 3
   Evaluate lim(x→3) (x² − 9)/(x − 3) by factoring. (Ans: 6)

3 Mark Questions

1. 1
   Evaluate lim(x→0) sin 2x / x. (Ans: 2)
2. 2
   Derive d/dx(x³) from first principles. (Ans: 3x²)
3. 3
   Evaluate lim(x→0) (sin ax)/(sin bx). (Ans: a/b)

4 Mark Questions

1. 1
   Using the first principle f′(x) = lim(h→0) [f(x+h) − f(x)]/h, derive the derivative of x² and hence of any polynomial xⁿ for n a positive integer. (Ans: 2x and n xⁿ⁻¹)
2. 2
   Evaluate lim(x→0) (1 − cos x)/(x²) and lim(x→0) (sin² x)/(x²), showing the standard limits used. (Ans: 1/2 and 1)