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Class 11 RD Sharma Solutions – Chapter 29 Limits – Exercise 29.4 | Set 2

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  • Last Updated : 28 Apr, 2021

Evaluates the following limits:

Question 18. Limx→1{√(5x – 4) – √x}/(x3 – 1)

Solution:

We have, Limx→1{√(5x – 4) – √x}/(x3 – 1)

Find the limit of the given equation

When we put x = 1, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→1}\frac{\sqrt{(5x - 4)} - \sqrt{x}}{(x^3 - 1)} \times \frac{\sqrt{(5x - 4)} + \sqrt{x}}{\sqrt{(5x - 4)} + \sqrt{x}}

= Limx→1{(5x – 4) – x}/[{√(5x – 4) + √x}(x3 – 1)]

= Limx→1{4(x – 1)}/[{√(5x – 4) + √x}(x-1)(x2 + x + 1)]

= Limx→1(4)/[{√(5x – 4) + √x}(x2 + x + 1)]

Now put x = 1, we get

= 4/{(3)(√1 + √1)}

= 4/6

= 2/3

Question 19. Limx→2{√(1 + 4x) – √(5 + 2x)}/(x – 2)

Solution:

We have, Limx→2{√(1 + 4x) – √(5 + 2x)}/(x – 2)

Find the limit of the given equation

When we put x = 2, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→2}\frac{\sqrt{(1 + 4x)} - \sqrt{(5 + 2x)}}{(x - 2)} \times \frac{\sqrt{(1 + 4x)} + \sqrt{(5 + 2x)}}{\sqrt{(1 + 4x)} + \sqrt{(5 + 2x)}}

= Limx→2{√(1 + 4x) – √(5+2x)}/[(x – 2){√(1 + 4x) + √(5 + 2x)}]

= Limx→2{(1 + 4x) – (5 + 2x)}/[(x – 2){√(1 + 4x) + √(5 + 2x)}]

= Limx→2{2(x – 2)}/[(x – 2){√(1 + 4x) + √(5 + 2x)}]

= Limx→2(2)/{√(1 + 4x) + √(5 + 2x)}

Now put x = 2, we get

= 2/{√(1 + 8) + √(5 + 4)}

= 2/(3 + 3)

= 1/3

Question 20. Limx→1{√(3 + x) – √(5 – x)}/(x2 – 1)

Solution:

We have, Limx→1{√(3 + x) – √(5 – x)}/(x2 – 1)

Find the limit of the given equation

When we put x = 1, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→1}\frac{\sqrt{(3 + x)} - \sqrt{(5 - x)}}{(x^2 - 1)} \times \frac{\sqrt{(3 + x)} + \sqrt{(5 - x)}}{\sqrt{(3 + x)} + \sqrt{(5 - x)}}

= Limx→1{(3 + x) – (5 – x)}/[(x2 – 1){√(3 + x) + √(5 – x)}]

= Limx→1{2(x – 1)}/[(x – 1)(x + 1){√(3 + x) + √(5 – x)}]

= Limx→1(2)/[(x + 1){√(3 + x) + √(5 – x)}]

Now put x = 1, we get

= 2/{2(2 + 2)}

= 1/4

Question 21. Limx→0{√(1 + x2) – √(1 – x2)}/(x)

Solution:

We have, Limx→0{√(1 + x2) – √(1 – x2)}/(x)

Find the limit of the given equation

When we put x = 0, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→0}\frac{\sqrt{(1 + x^2)} - \sqrt{(1 - x^2)}}{x} \times \frac{\sqrt{(1 + x^2)} + \sqrt{(1 - x^2)}}{\sqrt{(1 + x^2)} + \sqrt{(1 - x^2)}}

= Limx→0{(1 + x2) – (1 – x2)}/[x{√(1 + x2) + √(1 – x2)}]

= Limx→0{(1 + x2) – (1 – x2)}/[x{√(1 + x2) + √(1 – x2)}]

= Limx→0(2x2/[x{√(1 + x2) + √(1 – x2)}]

= Limx→0(2x/{√(1 + x2) + √(1 – x2)}

Now put x = 0, we get

= 2 × 0/(√1 + √1)

= 0

Question 22. Limx→0{√(1 + x + x2) – √(x + 1)}/(2x2)

Solution:

We have, Limx→0{√(1 + x + x2) – √(x + 1)}/(2x2)

Find the limit of the given equation

When we put x = 0, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→0}\frac{\sqrt{(1 + x + x^2)} - \sqrt{(x + 1)}}{2x^2} \times \frac{\sqrt{(1 + x + x^2)} + \sqrt{(x + 1)}}{\sqrt{(1 + x + x^2)} + \sqrt{(x + 1)}}

= Limx→0{(1 + x + x2) – (x + 1)}/[2x2{√(1 + x + x2) – √(x + 1)}]

= Limx→0(x2)/[2x2{√(1 + x + x2) – √(x + 1)}]

= Limx→0(1)/[2{√(1 + x + x2) – √(x + 1)}]

Now put x = 0, we get

= 1/{2(√1 + √1)

= 1/4

Question 23. Limx→4{2 – √x}/(4 – x)

Solution:

We have, Limx→4{2 – √x}/(4 – x)

Find the limit of the given equation

When we put x = 4, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→4}\frac{2 - √x}{(4 - x)} \times \frac{2 + √x}{2 + √x}

= Limx→4{4 – x}/[(4 – x){2 + √x}]

= Limx→4{4 – x}/[(4 – x){2 + √x}]

= Limx→4(1)/{2 + √x}

Now put x = 4, we get

= 1/(2 + 2)

= 1/4

Question 24. Limx→a(x – a)/{√x – √a}

Solution:

We have, Limx→a(x – a)/{√x – √a}

Find the limit of the given equation

When we put x = a, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

=   Lim_{x→a}\frac{(x - a)}{√x - √a} \times  \frac{√x + √a}{√x + √a}

= Limx→a[(x – a){√x – √a}]/(x – a)

= Limx→a{√x + √a}

Now put x = a, we get

= √a + √a

= 2√a

Question 25. Limx→0{√(1 + 3x) – √(1 – 3x)}/(x)

Solution:

We have, Limx→0{√(1 + 3x) – √(1 – 3x)}/(x)

Find the limit of the given equation

When we put x = a, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→0}\frac{\sqrt{(1 + 3x)} - \sqrt{(1 - 3x)}}{x} \times \frac{\sqrt{(1 + 3x)} + \sqrt{(1 - 3x)}}{\sqrt{(1 + 3x)} + \sqrt{(1 - 3x)}}

= Limx→0{(1 + 3x) – (1 – 3x)}/[(x){√(1 + 3x) – √(1 – 3x)}]

= Limx→0(6x)/[(x){√(1 + 3x) – √(1 – 3x)}]

= Limx→0(6)/{√(1 + 3x) – √(1 – 3x)}

Now put x = 0, we get

= 6/(√1 + √1)

= 6/2

= 3

Question 26. Limx→0{√(2 – x) – √(2 + x)}/(x)

Solution:

We have, Limx→0{√(2 – x) – √(2 + x)}/(x)

Find the limit of the given equation

When we put x = 0, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

 Lim_{x→0}\frac{\sqrt{(2 - x)} - \sqrt{(2 + x)}}{x} \times \frac{\sqrt{(2 - x)} + \sqrt{(2 + x)}}{\sqrt{(2 - x)} + \sqrt{(2 + x)}}

= Limx→0{(2 – x) – (2 + x)}/[x{√(2 – x) + √(2 + x)}]

= Limx→0(-2x)/[x{√(2 – x) + √(2 + x)}]

= Limx→0(-2)/{√(2 – x) + √(2 + x)}

Now put x = 0, we get

= (-2)/(√2 + √2)

= (-2)/(2√2)

= -1/(√2)

Question 27. Limx→1{√(3 + x) – √(5 – x)}/(x2 – 1)

Solution:

We have, Limx→1{√(3 + x) – √(5 – x)}/(x2 – 1)

Find the limit of the given equation

When we put x = 1, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{x→1}\frac{\sqrt{(3 + x)} - \sqrt{(5 - x)}}{(x^2 - 1)} \times \frac{\sqrt{(3 + x)} + \sqrt{(5 - x)}}{\sqrt{(3 + x)} + \sqrt{(5 - x)}}

= Limx→1{(3 + x) – (5 – x)}/[(x2 – 1){√(3 + x) + √(5 – x)}]

= Limx→1{2(x – 1)}/[(x – 1)(x + 1){√(3 + x) + √(5 – x)}]

= Limx→1(2)/[(x + 1){√(3 + x) + √(5 – x)}]

Now put x = 1, we get

= 2/{(2)(√4 + √4)}

= 2/8

= 1/4

Question 28. Limx→1{(2x – 3)(√x – 1)}/(3x2 + 3x – 6)

Solution:

We have, Limx→1{(2x – 3)(√x – 1)}/(3x2 + 3x – 6)

Find the limit of the given equation

When we put x = 1, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

= Limx→1{(2x – 3)(x – 1)}/[(3x2 + 3x – 6)(√x + 1)]

= Limx→1{(2x – 3)(x – 1)}/[3(x2 + x – 2)(√x + 1)]

= Limx→1{(2x – 3)(x – 1)}/[3(x – 1)(x + 2)(√x + 1)]

= Limx→1(2x – 3)/[3(x + 2)(√x + 1)]

Now put x = 1, we get

= (2 – 3)/{3(3)(√1 + 1)

= -1/(3 × 3 × 2)

= -1/18

Question 29. Limx→0{√(1 + x2) – √(1 + x)}/{√(1 + x3) – √(1 + x)}

Solution:

We have, Limx→0{√(1 + x2) – √(1 + x)}/{√(1 + x3) – √(1 + x)}

Find the limit of the given equation

When we put x = 0, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

=\lim_{x\to0}\frac{(\sqrt{1+x^2}-\sqrt{1+x})(\sqrt{1+x^2}+\sqrt{1+x})}{(\sqrt{1+x^3}-\sqrt{1+x})(\sqrt{1+x^3}+\sqrt{1+x})}

=\lim_{x\to0}\frac{x(x-1)}{x(x^2-1)}×\frac{(\sqrt{1+x^3}+\sqrt{x+1})}{(\sqrt{1+x^2}+\sqrt{x+1})}

=\lim_{x\to0}\frac{[(1+x^2)-(1+x)]}{[(1+x^3)-(1+x)}×\frac{(\sqrt{1+x^3}+\sqrt{x+1})}{(\sqrt{1+x^2}+\sqrt{x+1})}

=\lim_{x\to0}\frac{(x^2-x)}{(x^3-x)}×\frac{(\sqrt{1+x^3}+\sqrt{x+1})}{(\sqrt{1+x^2}+\sqrt{x+1})}

=\lim_{x\to0}\frac{1}{(x+1)}×\frac{(\sqrt{1+x^3}+\sqrt{x+1})}{(\sqrt{1+x^2}+\sqrt{x+1})}

Now put x = 0, we get

= (√1 + √1)/{1(√1 + √1)}

= 2/2

= 1

Question 30. Limx→1{x2 – √x}/{√x – 1}

Solution:

We have, Limx→1{x2 – √x}/{√x – 1}

Find the limit of the given equation

When we put x = 1, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

= Limx→1{√x(x√x -1)}/{√x – 1}

= Limx→1{√x(x3/2 – 1)}/{√x – 1}

= Limx→1[√x{(√x)3 – 1}]/{√x – 1}

= Limx→1[(√x)(√x – 1)(x + √x + 1)]/{√x – 1}

= Limx→1[(√x)(x + √x + 1)]

Now put x = 1, we get

= (√1)(1 + √1 + 1)

= 3

Question 31. Limh→0{√(x + h) – √x}/(h), x ≠ 0   

Solution:

We have, Limh→0{√(x + h) – √x}/(h)

Find the limit of the given equation

When we put h = 0, this expression takes the form of 0/0.

So, on rationalizing the given equation we get

Lim_{h→0}\frac{\sqrt{(x + h)} - \sqrt{x}}{h} \times \frac{\sqrt{(x + h)} + \sqrt{x}}{\sqrt{(x + h)} + \sqrt{x}}

= Limh→0{(x + h) – x}/[h{√(x + h) + √x}]

= Limh→0(h)/[h{√(x + h) + √x}]

= Limh→0(1)/{√(x + h) + √x}

Now put x = 0, we get

= 1/(√x + √x)

= 1/(2√x)

Question 32. Limx→√10{√(7 + 2x) – (√5 + √2)}/(x2 – 10)

Solution:

We have, Limx→√10{√(7 + 2x) – (√5 + √2)}/(x2 – 10)

= Limx→√10{√(7 + 2x) – √(√5 + √2)2}/{(x – √10)(x + √10)}

= Limx→√10{√(7 + 2x) – √(5 + 2 + 2√5√2)}/{(x – √10)(x + √10)}

= Limx→√10{√(7 + 2x) – √(7 + 2√10)}/{(x – √10)(x + √10)}

On rationalizing numerator.

=\lim_{x\to\sqrt{10}}\frac{(\sqrt{7+2x}-\sqrt{7+2\sqrt{10}})(\sqrt{7+2x}+\sqrt{7+2\sqrt{10}})}{(x-\sqrt{10})(x+\sqrt{10})(\sqrt{7+2x}+\sqrt{7+2\sqrt{10}})}

=\lim_{x\to\sqrt{10}}\frac{(7+2x)-(7+2\sqrt{10})}{(x-\sqrt{10})(x+\sqrt{10})(\sqrt{7+2x}+\sqrt{7+2\sqrt{10}})}

=\lim_{x\to\sqrt{10}}\frac{2(x-\sqrt{10})}{(x-\sqrt{10})(x+\sqrt{10})(\sqrt{7+2x}+\sqrt{7+2\sqrt{10}})}

Now put x = √10, we get

=\frac{2}{(\sqrt{10}+\sqrt{10})(2\sqrt{7+2\sqrt{10}}}

=\frac{1}{(2\sqrt{10})(\sqrt{7+2\sqrt{10}}}

=\frac{1}{(2\sqrt{10})(\sqrt{\sqrt{5}+\sqrt{2})^2}}

= 1/{(2√10)(√5 + √2)}

On rationalizing denominator.

= (√5 – √2)/{(2√10)(5 – 2)}

= (√5 – √2)/(6√10)

Question 33. Limx→√6{√(5 + 2x) – (√3 + √2)}/(x2 – 6)

Solution:

We have, Limx→√6{√(5 + 2x) – (√3 + √2)}/(x2 – 6)

= Limx→√6{√(5 + 2x) – √(√3 + √2)2}/{(x – √6)(x + √6)}

= Limx→√6{√(5 + 2x) – √(3 + 2 + 2√3√2)}/{(x – √6)(x + √6)}

= Limx→√6{√(5 + 2x) – √(5 + 2√6)}/{(x  -√6)(x + √6)}

On rationalizing numerator.

\lim_{x\to\sqrt{6}}\frac{(\sqrt{5+2x}-\sqrt{5+2\sqrt{6}})(\sqrt{5+2x}+\sqrt{5+2\sqrt{6}})}{(x-\sqrt{6})(x+\sqrt{6})(\sqrt{5+2x}+\sqrt{5+2\sqrt{6}})}

=\lim_{x\to\sqrt{6}}\frac{(5+2x)-(5+2\sqrt{6})}{(x-\sqrt{6})(x+\sqrt{6})(\sqrt{5+2x}+\sqrt{5+2\sqrt{6}})}

=\lim_{x\to\sqrt{6}}\frac{2(x-\sqrt{6})}{(x-\sqrt{6})(x+\sqrt{6})(\sqrt{5+2x}+\sqrt{5+2\sqrt{6}})}

=\lim_{x\to\sqrt{6}}\frac2{(x+\sqrt{6})(\sqrt{5+2x}+\sqrt{5+2\sqrt{6}})}

Now put x = √6, we get

=\frac2{(\sqrt6+\sqrt{6})(\sqrt{5+2\sqrt{5}}+\sqrt{5+2\sqrt{6}})}

=\frac2{(2\sqrt{6})(2\sqrt{5+2\sqrt{5}})}

=\frac1{(2\sqrt{6})(\sqrt{5+2\sqrt{5}})}

= 1/{(2√6)(√3 + √2)}

On rationalizing denominator, we get

= (√3 – √2)/{(2√6)(3 – 2)}

= (√3 – √2)/(2√6)

Question 34. Limx→√2{√(3 + 2x) – (√2 + 1)}/(x2 – 2)

Solution:

We have,  Limx→√2{√(3 + 2x) – (√2 + 1)}/(x2 – 2)

= Limx→√2{√(3 + 2x) – √(√2 + 1)2}/{(x – √2)(x + √2)}

= Limx→√2{√(3 + 2x) – √(2 + 1 + 2√3)}/{(x – √2)(x + √2)}

= Limx→√2{√(3 + 2x) – √(3 + 2√3)}/{(x – √2)(x + √2)}

On rationalizing numerator.

=\lim_{x\to\sqrt{2}}\frac{(\sqrt{3+2x}-\sqrt{3+2\sqrt{2}})(\sqrt{3+2x}+\sqrt{3+2\sqrt{2}})}{(x-\sqrt{2})(x+\sqrt{2})(\sqrt{3+2x}+\sqrt{3+2\sqrt{2}})}

=\lim_{x\to\sqrt{2}}\frac{(3+2x)-(3+2\sqrt2)}{(x-\sqrt{2})(x+\sqrt{2})(\sqrt{3+2x}+\sqrt{3+2\sqrt2})}

=\lim_{x\to\sqrt{2}}\frac{(3+2x)-(3+2\sqrt2)}{(x-\sqrt{2})(x+\sqrt{2})(\sqrt{3+2x}+\sqrt{3+2\sqrt2})}

=\lim_{x\to\sqrt{2}}\frac{2(x-\sqrt{3})}{(x-\sqrt{2})(x+\sqrt{2})(\sqrt{3+2x}+\sqrt{3+2\sqrt2})}

=\lim_{x\to\sqrt{2}}\frac{2}{(x+\sqrt{2})(\sqrt{3+2x}+\sqrt{3+2\sqrt2})}

Now put x = √2, we get

=\frac2{(\sqrt2+\sqrt2)(\sqrt{3+2\sqrt{5}}+\sqrt{3+2\sqrt2})}

=\frac2{(2\sqrt2)(2\sqrt{3+2\sqrt2})}

=\frac1{(2\sqrt{2})(\sqrt{3+2\sqrt2})}

= 1/{(2√2)(√2 + 1)}

On rationalizing denominator, we get

= (√2 – 1)/{(2√2)(2 – 1)}

= (√2 – 1)/(2√2)


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