Class 11 RD Sharma Solutions – Chapter 29 Limits – Exercise 29.5 | Set 1

Evaluate the following limits:

Question 1. $\lim_{x \to a} \frac {(x+2)^{\frac 5 2}-(a+2)^{\frac 5 2}} {x-a}$

Solution:

$\lim_{x \to a} \frac {(x+2)^{\frac 5 2}-(a+2)^{\frac 5 2}} {x-a}$
$=\lim_{x \to a} \frac {(x+2)^{\frac 5 2}-(a+2)^{\frac 5 2}} {(x+2)-(a+2)}$
Let y = x + 2 and b = a + 2
$=\lim_{y \to b} \frac {(y)^{\frac 5 2}-(b)^{\frac 5 2}} {(y)-(b)}$
$=\frac 5 2 b^{{\frac 5 2} -1}$ [using formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ ]
$=\frac 5 2 (a+2)^{{\frac 5 2} -1}$
$=\frac 5 2 (a+2)^{\frac 3 2}$

Question 2. $\lim_{x \to a} \frac {(x+2)^{\frac 3 2}-(a+2)^{\frac 3 2}} {x-a}$

Solution:

$\lim_{x \to a} \frac {(x+2)^{\frac 3 2}-(a+2)^{\frac 3 2}} {x-a}$
$=\lim_{x \to a} \frac {(x+2)^{\frac 3 2}-(a+2)^{\frac 3 2}} {(x+2)-(a+2)}$
Let y=x+2 and b=a+2
$=\lim_{y \to b} \frac {(y)^{\frac 3 2}-(b)^{\frac 3 2}} {(y)-(b)}$
$=\frac 3 2 b^{{\frac 3 2} -1}$ [using formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ ]
$=\frac 3 2 (a+2)^{{\frac 3 2} -1}$
$=\frac 3 2 (a+2)^{\frac 1 2}$

Question 3. $\lim_{x \to 0} \frac {(1+x)^{6}-1} {(1+x)^2-1}$

Solution:

$\lim_{x \to 0} \frac {(1+x)^{6}-1} {(1+x)^2-1}$
Dividing the numerator and denominator with 1+x-1
$=\lim_{x \to 0} \frac {\frac {(1+x)^{6}-1^6} {1+x-1} } {\frac {(1+x)^{2}-1^2} {1+x-1}}$
Let y=1+x as $x \to 0,y\to 1$
$=\lim_{y \to 1} \frac {\frac {y^{6}-1^6} {y-1} } {\frac {y^{2}-1^2} {y-1}}$
$=\frac {6(1)^{6-1}} {2(1)^{2-1}}$ [using formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ ]
$=\frac 6 2$
$=3$

Question 4. $\lim_{x \to a} \frac {x^{\frac 2 7}-a^{\frac 2 7}} {x-a}$

Solution:

$\lim_{x \to a} \frac {x^{\frac 2 7}-a^{\frac 2 7}} {x-a}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ , here, $n=\frac 2 7$
$=\frac 2 7a^{{\frac 2 7}-1}$
$=\frac 2 7 a^{\frac {-5} {7}}$

Question 5. $\lim_{x \to a} \frac {x^{\frac 5 7}-a^{\frac 5 7}} {x^{\frac 2 7}-a^{\frac 2 7}}$

Solution:

$\lim_{x \to a} \frac {x^{\frac 5 7}-a^{\frac 5 7}} {x^{\frac 2 7}-a^{\frac 2 7}}$
Dividing the numerator and denominator with x-a
$=\lim_{x \to a} \frac { \frac {x^{\frac 5 7}-a^{\frac 5 7}} {x-a}} {\frac {x^{\frac 2 7}-a^{\frac 2 7}} {x-a}}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ , here, $n=\frac 5 7$ in numerator and $n=\frac 2 7$ in denominator
$=\frac {\frac 5 7a^{{\frac 5 7}-1}} {\frac 2 7a^{{\frac 2 7}-1}}$
$=\frac {\frac 5 7a^{\frac {-2} 7}} {\frac 2 7a^{\frac {-5} 7}}$
$=\frac 5 2a^{\frac {-2} {7} + \frac {5} {7}}$
$=\frac 5 2a^{\frac {3} {7}}$

Question 6. $\lim_{x \to {\frac {-1} 2}} \frac {8x^3+1} {2x+1}$

Solution:

$\lim_{x \to {\frac {-1} 2}} \frac {8x^3+1} {2x+1}$
$=\frac 8 2\lim_{x \to {\frac {-1} 2}} \frac {x^3+{(\frac 1 2)}^3} {x+\frac 1 2}$
$=4\lim_{x \to {\frac {-1} 2}} \frac {x^3-{(-\frac 1 2)}^3} {x-(-\frac 1 2)}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ here, $n=3$ and $a=\frac {-1} {2}$
$=4 \times 3(\frac {-1} 2)^{3-1}$
$=4 \times 3 \times \frac 1 4$
$=3$

Question 7. $\lim_{x \to 27} \frac {({x^{\frac 1 3}+3})({x^{\frac 1 3}-3})} {x-27}$

Solution:

$\lim_{x \to 27} \frac {({x^{\frac 1 3}+3})({x^{\frac 1 3}-3})} {x-27}$
$=\lim_{x \to 27} \frac {({x^{\frac 2 3}-9})} {x-27}$
$=\lim_{x \to 27} \frac {({x^{\frac 2 3}-27^{\frac 2 3}})} {x-27}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$
$=\frac 2 3(27)^{\frac 2 3 -1}$
$=\frac 2 3(27)^{\frac {-1} 3}$
$=\frac 2 3\times \frac 1 {(27)^{\frac 1 3}}$
$=\frac 2 3\times \frac 1 3$
$=\frac 2 9$

Question 8. $\lim_{x \to 4} \frac {{x^3-64}} {x^2-16}$

Solution:

$\lim_{x \to 4} \frac {{x^3-64}} {x^2-16}$
$=\lim_{x \to 4} \frac {{x^3-4^3}} {x^2-4^2}$
Dividing the numerator and denominator with x-4
$=\lim_{x \to 4} \frac {\frac {x^3-4^3} {x-4}} {\frac {x^2-4^2} {x-4}}$
$= \frac {\lim_{x \to 4}{\frac {x^3-4^3} {x-4}}} {\lim_{x \to 4}{\frac {x^2-4^2} {x-4}}}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ here, n=3 in numerator and n=2 in denominator
$= \frac {3(4)^{3-1}} {2(4)^{2-1}}$
$= \frac {3(4)^{2}} {2(4)}$
$=6$

Question 9. $\lim_{x \to 1} \frac {{x^{15}-1}} {x^{10}-1}$

Solution

$\lim_{x \to 1} \frac {{x^{15}-1}} {x^{10}-1}$
Dividing the numerator and denominator with x-1
$=\lim_{x \to 1} \frac {\frac {x^{15}-1^{15}} {x-1}} {\frac {x^{10}-1^{10}} {x-1}}$
$= \frac {\lim_{x \to 1}\frac {x^{15}-1^{15}} {x-1}} {\lim_{x \to 1}\frac {x^{10}-1^{10}} {x-1}}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$ here,
n=15 in numerator and n=10 in denominator
$=\frac {15(1)^{15-1}}{10(1)^{10-1}}$
$=\frac {15} {10}$
$= \frac 3 2$

Question 10. $\lim_{x \to {-1}} \frac {{x^{3}+1}} {x+1}$

Solution:

$\lim_{x \to {-1}} \frac {{x^{3}+1}} {x+1}$
$=\lim_{x \to {-1}} \frac {{x^{3}-(-1)^3}} {x-(-1)}$
Applying the formula $\lim_{x \to a} \frac {x^{n}-a^{n}} {x-a}=na^{n-1}$
$=3(-1)^{3-1}$
$=3(-1)^2$
$=3$

Last Updated : 03 Mar, 2021

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 29.5 | Set 1

Evaluate the following limits:

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

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Question 1. $\lim_{x \to a} \frac {(x+2)^{\frac 5 2}-(a+2)^{\frac 5 2}} {x-a}$

Question 2. $\lim_{x \to a} \frac {(x+2)^{\frac 3 2}-(a+2)^{\frac 3 2}} {x-a}$

Question 3. $\lim_{x \to 0} \frac {(1+x)^{6}-1} {(1+x)^2-1}$

Question 4. $\lim_{x \to a} \frac {x^{\frac 2 7}-a^{\frac 2 7}} {x-a}$

Question 5. $\lim_{x \to a} \frac {x^{\frac 5 7}-a^{\frac 5 7}} {x^{\frac 2 7}-a^{\frac 2 7}}$

Question 6. $\lim_{x \to {\frac {-1} 2}} \frac {8x^3+1} {2x+1}$

Question 7. $\lim_{x \to 27} \frac {({x^{\frac 1 3}+3})({x^{\frac 1 3}-3})} {x-27}$

Question 8. $\lim_{x \to 4} \frac {{x^3-64}} {x^2-16}$

Question 9. $\lim_{x \to 1} \frac {{x^{15}-1}} {x^{10}-1}$

Question 10. $\lim_{x \to {-1}} \frac {{x^{3}+1}} {x+1}$