Class 11 RD Sharma Solutions – Chapter 29 Limits – Exercise 29.6 | Set 2

Question 14. Lim_n→∞{1²+ 2²+ ……………. + n²}/(n³)

Solution:

We have,
Lim_n→∞{1²+ 2²+ ……………. + n²}/(n³)
= Lim_n→∞[n(n+1)(2n+1)]/6n³
= Lim_n→∞[(n+1)(2n+1)]/6n²
= $\lim_{n\to∞}\frac{(n+1)}{n}\frac{2n+1}{n}\frac{1}{6}$
$\lim_{n\to∞}(1+\frac{1}{n})(2+\frac{1}{n})\frac{1}{6}$
When n → ∞, (1/n) → 0
= 2/6
= 1/3

Question 15. Lim_n→∞{1 + 2 + 3 + 4 +……………. + n – 1}/n²

Solution:

We have,
Lim_n→∞{1 + 2 + 3 + 4 +……………. + n – 1}/n²
= Lim_n→∞[n(n – 1)/2n²]
= Lim_n→∞[n²– n/2n²]
= Lim_n→∞(1/2 – 1/2n)
When n → ∞, (1/n) → 0
= 1/2

Question 16. Lim_n→∞{1³+ 2³+ ……………. + n³}/(n⁴)

Solution:

We have,
Lim_n→∞{1³+ 2³+ ……………. + n³}/(n⁴)
= Lim_n→∞[n²(n + 1)²]/(4n⁴) [since (1³+ 2³+ …………. + n³) = n²(n + 1)²/4]
= Lim_n→∞[(n + 1)²]/(4n²)
= Lim_n→∞[(1 + 1/n)²× (1/4)]
When n → ∞, (1/n) → 0
= 1/4

Question 17. Lim_n→∞{1³+ 2³+ ……………. + n³}/(n – 1)⁴

Solution:

We have,
Lim_n→∞{1³+ 2³+ ……………. + n³}/(n – 1)⁴
= Lim_n→∞[n²(n + 1)²]/[4(n – 1)⁴] [since (1³+ 2³+ …………. + n³) = n²(n + 1)²/4]
= $\lim_{n\to∞}\frac{n^4(1+\frac{1}{n})^2}{4n^4(1-\frac{1}{n})^4}$
= $\lim_{n\to∞}\frac{(1+\frac{1}{n})^2}{4(1-\frac{1}{n})^4}$
When n → ∞, (1/n) → 0
= 1/4

Question 18. Lim_x→∞[√x{√(x + 1) – √x}]

Solution:

We have,
Lim_x→∞[√x{√(x + 1) – √x}]
On rationalizing numerator, we get
= Lim_x→∞[(√x){(x + 1) – x}]/{√(x + 1) + √x}
= Lim_x→∞(√x}/{√(x + 1) + √x}
= $\lim_{x\to∞}\frac{\sqrt{x}}{\sqrt{x}(1+\frac{1}{x}+1)}$
When x → ∞, (1/x) → 0.
= 1/(√1 + 1)
= 1/2

Question 19. Lim_x→∞[1/3 + 1/3²+ 1/3³+ ……………… + 1/3ⁿ]

Solution:

We have,
Lim_x→∞[1/3 + 1/3²+ 1/3³+ ……………… + 1/3ⁿ]
This is G.P series of common ratio 1/3.
So, the sum of n terms of G.P. S_n= [a(1 – rⁿ)]/(1 – r) (i)
a = 1/3, r = 1/3
On putting the value of a & r in equation (i), we get
S_n= (1/2)(1 – 1/3ⁿ)
= Lim_x→∞[(1/2)(1 – 1/3ⁿ)]
= (1/2)Lim_x→∞(1 – 1/3ⁿ)
= (1/2)(1 – 0)
= 1/2

Question 20. Lim_x→∞{(x⁴+ 7x³+ 46x + a)}/{(x⁴+ 6)}.

Solution:

We have,
Lim_x→∞{(x⁴+ 7x³+ 46x + a)}/{(x⁴+ 6)}
= $\lim_{x\to∞}\frac{x^4(1+\frac{7}{x}+\frac{46}{x^2}+\frac{a}{x^4})}{x^4(1+\frac{6}{x^4})}$
When x → ∞, (1/x), (1/x²), (1/x³), (1/x⁴) → 0
= 1/1
= 1

Question 21. f(x) = (ax²+ b)/(x²+ 1), Lim_x→0f(x) = 1, Lim_x→∞f(x) = 1, then prove that f(-2) = f(2) = 1

Solution:

We have,
f(x) = (ax²+ b)/(x²+ 1)
= Lim_x→0[(ax²+ b)/(x²+ 1)]
b/1 = 1
b = 1
= Lim_x→∞[(ax²+ b)/(x²+ 1)]
= $\lim_{x\to∞}\frac{x^2(a+\frac{b}{x^2})}{x^2(1+\frac{1}{x^2})}=1$
When x → ∞, (1/x²) → 0.
(a + 0)/(1 + 0) = 1
a = 1
Hence, a = 1, b = 1
f(x) = (x²+ 1)/(x²+ 1)
f(x) = 1
f(-2) = 1
f(2) = 1 (Since f(x) is independent on x)
f(-2) = f(2) = 1
Hence proved

Question 22. Show that Lim_x→∞[√(x²+ x + 1) – x] ≠ Lim_x→∞[√(x²+ 1) – x]

Solution:

We have,
L.H.S,
= Lim_x→∞[√(x²+ x + 1) – x]
On rationalizing numerator, we get
= Lim_x→∞[(x²+ x + 1) – x²]/[√(x²+ x + 1) + x]
= Lim_x→∞(x + 1)/[√(x²+ x + 1) + x]
= Lim_x→∞[x(1 + 1/x)/[x{√(1 + 1/x + 1/x²) + 1}]
= Lim_x→∞[(1 + 1/x)/[{√(1 + 1/x + 1/x²) + 1}]
When x → ∞, (1/x), (1/x²) → 0.
= 1/(√1 + 1)
= 1/2
Now we solve R.H.S,
= Lim_x→∞[√(x²+ 1) – x]
On rationalizing numerator, we get
= Lim_x→∞[(x²+ 1) – x²]/[√(x²+ 1) + x]
= Lim_x→∞(1)/[√(x²+ 1) + x]
= 1/[√(∞ + 1) + ∞]
= 1/∞
= 0
L.H.S ≠ R.H.S
Hence, Lim_x→∞[√(x²+ x + 1) – x] ≠ Lim_x→∞[√(x²+ 1) – x]

Question 23. Lim_x→-∞[√(4x²– 7x) + 2x]

Solution:

We have,
Lim_x→-∞[√(4x²– 7x) + 2x]
Let x = -n when x → -∞, then n → ∞.
= Lim_n→∞[√(4n²+ 7n) – 2n]
On rationalizing numerator, we get
= Lim_n→∞[(4n²+ 7n) – 4n²]/[√(4n²+ 7n) + 2n]
= Lim_n→∞[(7n)/[√(4n²+ 7n) + 2n]
= Lim_n→∞(7n)/[n{√(4 + 7/n) + 2}]
= Lim_n→∞(7)/{√(4 + 7/n) + 2}
When n → ∞, (1/n) → 0
= 7/(√4 + 2)
= 7/(2 + 2)
= 7/4

Question 24. Lim_x→-∞[√(x²– 8x) + x]

Solution:

We have,
Lim_x→-∞[√(x²– 8x) + x]
Let x = -n when x → -∞, then n → ∞.
= Lim_n→∞[√(n²+ 8n) – n]
On rationalizing numerator, we get
= Lim_n→∞[(n²+ 8n) – n²]/[√(n²+ 8n) + n]
= Lim_n→∞[(8n)/[√(n²+ 8n) + n]
= Lim_n→∞(8n)/[n{√(1 + 8/n) + 1}]
= Lim_n→∞(8)/{√(1 + 8/n) + 1}
When n → ∞, (1/n) → 0
= 8/(√1 + 1)
= 8/2
= 4

Question 25. Lim_n→∞(1⁴+ 2⁴+ ……….+ n⁴)/n⁵– Lim_n→∞(1³+ 2³+ ………. + n³)/n⁵

Solution:

We have,
Lim_n→∞(1⁴+ 2⁴+ ……….+ n⁴)/n⁵– Lim_n→∞(1³+ 2³+ ………. + n³)/n⁵
= $\lim_{n\to∞}\frac{\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}}{n^5}-\lim_{n\to∞}\frac{\frac{n^2(n+1)^2}{4}}{n^5}$
= $\lim_{n\to∞}\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30n^5}-\lim_{n\to∞}\frac{n^2(n+1)^2}{4n^5}$
= $\lim_{n\to∞}\frac{(n+1)(2n+1)(3n^2+3n-1)}{30n^4}-\lim_{n\to∞}\frac{(n+1)^2}{4n^3}$
= $\frac{1}{30}\lim_{n\to∞}(1+\frac{1}{n})(2+\frac{1}{n})(3+\frac{3}{n}-\frac{1}{n^2})-\frac{1}{4}\lim_{n\to∞}\frac{1}{n}(1+\frac{1}{n})^2$
When n → ∞, (1/n), (1/n²), (1/n³) → 0
= 1/3 × 1 × 2 × 3 – 1/4 × 0
= 6/30
= 1/5

Question 26. Lim_n→∞{(1.2 + 2.3 + 3.4 + ……….+ n (n + 1)}/n³

Solution:

We have,
Lim_n→∞{(1.2 + 2.3 + 3.4 + ……….+ n (n + 1)}/n³
= $\lim_{n\to∞}\frac{\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}}{n^3}$
= $\lim_{n\to∞}\frac{n(n+1)[\frac{(2n+1)+3}{6}]}{n^3}$
= $\lim_{n\to∞}\frac{n(n+1)[\frac{(2n+4)}{6}]}{n^3}$
= $\lim_{n\to∞}\frac{n(n+1)[(2n+4)]}{6n^3}$
= $\lim_{n\to∞}\frac{n(n+1)[(2n+4)]}{6n^3}$
= $\lim_{n\to∞}\frac{(1+\frac{1}{n})(2+\frac{4}{n})}{6}$
When n → ∞, (1/n) → 0
= (1 × 2)/6
= 2/6
= 1/3

Last Updated : 18 Apr, 2022

Chapter 1: Sets

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Trigonometric Ratios of Compound Angles

Chapter 8: Transformation Formulae

Chapter 9: Trigonometric Ratios of Multiple and Sub Multiple Angles

Chapter 10: Sine and Cosine Formulae and their Applications

Chapter 11: Trigonometric Equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progressions

Chapter 20: Geometric Progressions

Chapter 21: Some Special Series

Chapter 22: Brief Review of Cartesian System of Rectangular Coordinates

Chapter 23: The Straight Lines

Chapter 24: The Circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to 3D Coordinate Geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical Reasoning

Chapter 32: Statistics

Chapter 33: Probability

Chapter 1: Sets

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Trigonometric Ratios of Compound Angles

Chapter 8: Transformation Formulae

Chapter 9: Trigonometric Ratios of Multiple and Sub Multiple Angles

Chapter 10: Sine and Cosine Formulae and their Applications

Chapter 11: Trigonometric Equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progressions

Chapter 20: Geometric Progressions

Chapter 21: Some Special Series

Chapter 22: Brief Review of Cartesian System of Rectangular Coordinates

Chapter 23: The Straight Lines

Chapter 24: The Circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to 3D Coordinate Geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical Reasoning

Chapter 32: Statistics

Chapter 33: Probability

Class 11 RD Sharma Solutions – Chapter 29 Limits – Exercise 29.6 | Set 2

Question 14. Limn→∞{12 + 22 + ……………. + n2}/(n3)

Question 15. Limn→∞{1 + 2 + 3 + 4 +……………. + n – 1}/n2

Question 16. Limn→∞{13 + 23 + ……………. + n3}/(n4)

Question 17. Limn→∞{13 + 23 + ……………. + n3}/(n – 1)4

Question 18. Limx→∞[√x{√(x + 1) – √x}]

Question 19. Limx→∞[1/3 + 1/32 + 1/33 + ……………… + 1/3n]

Question 20. Limx→∞{(x4 + 7x3 + 46x + a)}/{(x4 + 6)}.

Question 21. f(x) = (ax2 + b)/(x2 + 1), Limx→0f(x) = 1, Limx→∞f(x) = 1, then prove that f(-2) = f(2) = 1

Question 22. Show that Limx→∞[√(x2 + x + 1) – x] ≠ Limx→∞[√(x2 + 1) – x]

Question 23. Limx→-∞[√(4x2 – 7x) + 2x]

Question 24. Limx→-∞[√(x2 – 8x) + x]

Question 25. Limn→∞(14 + 24 + ……….+ n4)/n5 – Limn→∞(13 + 23 + ………. + n3)/n5

Question 26. Limn→∞{(1.2 + 2.3 + 3.4 + ……….+ n (n + 1)}/n3

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Question 14. Lim_n→∞{1²+ 2²+ ……………. + n²}/(n³)

Question 15. Lim_n→∞{1 + 2 + 3 + 4 +……………. + n – 1}/n²

Question 16. Lim_n→∞{1³+ 2³+ ……………. + n³}/(n⁴)

Question 17. Lim_n→∞{1³+ 2³+ ……………. + n³}/(n – 1)⁴

Question 18. Lim_x→∞[√x{√(x + 1) – √x}]

Question 19. Lim_x→∞[1/3 + 1/3²+ 1/3³+ ……………… + 1/3ⁿ]

Question 20. Lim_x→∞{(x⁴+ 7x³+ 46x + a)}/{(x⁴+ 6)}.

Question 21. f(x) = (ax²+ b)/(x²+ 1), Lim_x→0f(x) = 1, Lim_x→∞f(x) = 1, then prove that f(-2) = f(2) = 1

Question 22. Show that Lim_x→∞[√(x²+ x + 1) – x] ≠ Lim_x→∞[√(x²+ 1) – x]

Question 23. Lim_x→-∞[√(4x²– 7x) + 2x]

Question 24. Lim_x→-∞[√(x²– 8x) + x]

Question 25. Lim_n→∞(1⁴+ 2⁴+ ……….+ n⁴)/n⁵– Lim_n→∞(1³+ 2³+ ………. + n³)/n⁵

Question 26. Lim_n→∞{(1.2 + 2.3 + 3.4 + ……….+ n (n + 1)}/n³