Class 11 RD Sharma Solutions - Chapter 29 Limits - Exercise 29.10 | Set 1

Evaluate the following limits:

Question 1. \lim_{x\to0}\frac{5^x-1}{\sqrt{4+x}-2}

Solution:

We have,

=\lim_{x\to0}\frac{5^x-1}{\sqrt{4+x}-2}

=\lim_{x\to0}\frac{(5^x-1)(\sqrt{4+x}+2)}{(\sqrt{4+x}-2)(\sqrt{4+x}+2)}

=\lim_{x\to0}\frac{(5^x-1)(\sqrt{4+x}+2)}{4+x-4}

=\lim_{x\to0}\frac{(5^x-1)(\sqrt{4+x}+2)}{x}

=\lim_{x\to0}\frac{(5^x-1)}{x}\lim_{x\to0}(\sqrt{4+x}+2)

=4\lim_{x\to0}\frac{(5^x-1)}{x}

We know,lim_{x\to0}\frac{a^x-1}{x}=loga . So, we have,

= 4 log 5

Question 2. \lim_{x\to0}\frac{log(1+x)}{3^x-1}

Solution:

We have,

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

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

We know,lim_{x\to0}\frac{log(1+x)}{x}=1 . So, we have,

=\frac{1}{\lim_{x\to0}\frac{3^x-1}{x}}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

=\frac{1}{log3}

Question 3. \lim_{x\to0}\frac{a^x+a^{-x}-2}{x^2}

Solution:

We have,

=\lim_{x\to0}\frac{a^x+a^{-x}-2}{x^2}

=\lim_{x\to0}\frac{a^{2x}+1-2a^x}{x^2}

=\lim_{x\to0}(\frac{a^x-1}{x})^2×\frac{1}{a^x}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

=(log_ea)^2×\frac{1}{a^0}

= (log_e a)²

Question 4. \lim_{x\to0}\frac{a^{mx}-1}{b^{nx}-1} , n≠0

Solution:

We have,

=\lim_{x\to0}\frac{a^{mx}-1}{b^{nx}-1}

=\lim_{x\to0}\frac{a^{mx}-1}{mx}×\frac{1}{\lim_{x\to0}\frac{b^{nx}-1}{nx}}×\frac{m}{n}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

=loga×\frac{1}{logb}×\frac{m}{n}

=\frac{mloga}{nlogb}

Question 5. \lim_{x\to0}\frac{a^x+b^x-2}{x}

Solution:

We have,

=\lim_{x\to0}\frac{a^x+b^x-2}{x}

=\lim_{x\to0}\frac{(a^x-1)+(b^x-1)}{x}

=\lim_{x\to0}\left(\frac{a^x-1}{x}+\frac{b^x-1}{x}\right)

=\lim_{x\to0}\frac{a^x-1}{x}+\lim_{x\to0}\frac{b^x-1}{x}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= log a + log b

= log (ab)

Question 6. lim_{x\to0}(\frac{9^x-2.6^x+4^x}{x^2})

Solution:

We have,

=lim_{x\to0}(\frac{9^x-2.6^x+4^x}{x^2})

=lim_{x\to0}\frac{(3^x)^2-2.6^x+(2^x)^2}{x^2}

=lim_{x\to0}\left(\frac{3^x-2^x}{x}\right)^2

=lim_{x\to0}\left(\frac{3^x-1-(2^x-1)}{x}\right)^2

=lim_{x\to0}\left(\frac{3^x-1}{x}-\frac{2^x-1}{x}\right)^2

=\left(lim_{x\to0}\frac{3^x-1}{x}-lim_{x\to0}\frac{2^x-1}{x}\right)^2

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= (log 3 − log 2)²

=\left(log\frac{3}{2}\right)^2

Question 7. \lim_{x\to0}\frac{8^x-4^x-2^x+1}{x^2}

Solution:

We have,

=\lim_{x\to0}\frac{8^x-4^x-2^x+1}{x^2}

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

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

=2\lim_{x\to0}\left(\frac{2^x-1}{x}\right)^2

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= 2(log2)²

Question 8. \lim_{x\to0}\frac{a^{mx}-b^{nx}}{x}

Solution:

We have,

=\lim_{x\to0}\frac{a^{mx}-b^{nx}}{x}

=\lim_{x\to0}\frac{(a^{mx}-1)-(b^{nx}-1)}{x}

=\lim_{x\to0}\left(\frac{a^{mx}-1}{x}-\frac{b^{nx}-1}{x}\right)

=\lim_{x\to0}\left(m×\frac{a^{mx}-1}{mx}\right)-\lim_{x\to0}\left(n×\frac{b^{nx}-1}{nx}\right)

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= m log a − n log b

= log a^m − log bⁿ

=log\left(\frac{a^m}{b^n}\right)

Question 9. \lim_{x\to0}\frac{a^x+b^x+c^x-3}{x}

Solution:

We have,

=\lim_{x\to0}\frac{a^x+b^x+c^x-3}{x}

=\lim_{x\to0}\frac{(a^x-1)+(b^x-1)+(c^x-1)}{x}

=\lim_{x\to0}\frac{a^x-1}{x}+\lim_{x\to0}\frac{b^x-1}{x}+\lim_{x\to0}\frac{c^x-1}{x}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= log a + log b + log c

= log (abc)

Question 10. \lim_{x\to2}\frac{x-2}{log_a(x-1)}

Solution:

We have,

=\lim_{x\to2}\frac{x-2}{log_a(x-1)}

Let x − 2 = h. So, we get,

=\lim_{h\to0}\frac{h}{log_a(h+1)}

=\lim_{h\to0}\frac{h}{\frac{log(h+1)}{loga}}

=\lim_{h\to0}\frac{loga}{\frac{log(h+1)}{h}}

=\frac{loga}{\lim_{h\to0}\frac{log(h+1)}{h}}

We know,lim_{x\to{a}}\frac{log(1+x)}{x}=1 . So, we have,

= log a

Question 11. \lim_{x\to0}\frac{5^x+3^x+2^x-3}{x}

Solution:

We have,

=\lim_{x\to0}\frac{5^x+3^x+2^x-3}{x}

=\lim_{x\to0}\frac{(5^x-1)+(3^x-1)+(2^x-1)}{x}

=\lim_{x\to0}\frac{5^x-1}{x}+\lim_{x\to0}\frac{4^x-1}{x}+\lim_{x\to0}\frac{3^x-1}{x}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= log 5 + log 3 + log 2

= log (5×3×2)

= log 30

Question 12. \lim_{x\to0}(a^{\frac{1}{x}}-1)x

Solution:

We have,

=\lim_{x\to0}(a^{\frac{1}{x}}-1)x

Let 1/x = h. We get,

=\lim_{h\to0}(a^h-1)\frac{1}{h}

=\lim_{h\to0}\frac{a^h-1}{h}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= log a

Question 13. \lim_{x\to0}\frac{a^{mx}-b^{nx}}{sinkx}

Solution:

We have,

=\lim_{x\to0}\frac{a^{mx}-b^{nx}}{sinkx}

=\lim_{x\to0}\frac{a^{mx}-b^{nx}}{kx×\frac{sinkx}{kx}}

=\frac{1}{k}\lim_{x\to0}\frac{\frac{a^{mx}-b^{nx}}{x}}{\frac{sinkx}{kx}}

=\frac{1}{k}\lim_{x\to0}\frac{\frac{a^{mx}-1-(b^{nx}-1)}{x}}{\frac{sinkx}{kx}}

=\frac{1}{k}\lim_{x\to0}\frac{\frac{a^{mx}-1}{x}-\frac{b^{nx}-1}{x}}{\frac{sinkx}{kx}}

=\frac{1}{k}\lim_{x\to0}\frac{m×\frac{a^{mx}-1}{mx}-n×\frac{b^{nx}-1}{nx}}{\frac{sinkx}{kx}}

Aslim_{x\to0}\frac{a^x-1}{x}=loga andlim_{x\to0}\frac{sinx}{x}=1 , we get,

=\frac{mloga-nlogb}{k}

=\frac{loga^m-logb^n}{k}

=\frac{1}{k}log(\frac{a^m}{b^n})

Question 14. \lim_{x\to0}\frac{a^x+b^x-c^x-d^x}{x}

Solution:

We have,

=\lim_{x\to0}\frac{a^x+b^x-c^x-d^x}{x}

=\lim_{x\to0}\frac{(a^x-1)+(b^x-1)-(c^x-1)-(d^x-1)}{x}

=\lim_{x\to0}\frac{a^x-1}{x}+\lim_{x\to0}\frac{b^x-1}{x}-\lim_{x\to0}\frac{c^x-1}{x}-\lim_{x\to0}\frac{d^x-1}{x}

Aslim_{x\to0}\frac{a^x-1}{x}=loga , we get,

= log a + log b − log c − log d

= log a+ log b − (log c + log d)

= log ab − log cd

=\frac{ab}{cd}

Question 15. \lim_{x\to0}\frac{e^x-1+sinx}{x}

Solution:

We have,

=\lim_{x\to0}\frac{e^x-1+sinx}{x}

=\lim_{x\to0}\frac{e^x-1}{x}+\lim_{x\to0}\frac{sinx}{x}

Aslim_{x\to0}\frac{a^x-1}{x}=loga andlim_{x\to0}\frac{sinx}{x}=1 , we get,

= log e + 1

= 1 + 1

= 2

Summary

Exercise 29.10 Set 1 covers evaluating limits of various functions, including algebraic, trigonometric, and exponential functions. Students learn to apply limit rules and techniques, such as the product rule, quotient rule, and chain rule. Limits measure the behavior of functions as the input changes. Understanding limits is crucial for calculus and its applications. Practice questions reinforce learning and application. Limits help model real-world phenomena and make predictions.