Exercise 27.3
Page no 27.48
Type 1
Question 1
Evaluate the following limits :
1. $\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{2 x}}{x}$
2. $\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{x}$
3. $\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}$
4. $\lim _{x \rightarrow 0} \frac{2^{x}-1}{x}$
5. $\lim _{x \rightarrow 0} \frac{3^{x}-2^{x}}{x}$
6. $\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}$
7. $\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}$
8. $\lim _{x \rightarrow 0} \frac{e^{2+x}-e^{2}}{x}$
9. $\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}$
Page no 27.49
(10. (i) $\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{\sin x}$
(ii) $\lim _{x \rightarrow 0} \frac{e^{x}-\sin x-1}{x}$
11. $\lim _{x \rightarrow 5} \frac{e^{x}-e^{5}}{x-5}$
12. $\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{x-5}$
13. $\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x}$
14. $\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}$
15. $\lim _{x \rightarrow 0} \frac{e^{h x}-e^{a x}}{x}$, where $0<a<b$
16. $\lim _{x \rightarrow 3} \frac{e^{x}-e^{3}}{x-3}$
17. $\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\cos x}-1}{\cos x}$
18. $\lim _{x \rightarrow 0} \frac{x\left(e^{2+x}-e^{2}\right)}{1-\cos x}$
19. $\lim _{x \rightarrow 0} \frac{x\left(e^{x}-1\right)}{1-\cos x}$
20. $\lim _{x \rightarrow 0} \frac{x 3^{x}-x}{1-\cos x}$
21. $\lim _{h \rightarrow 0} \frac{e^{x+h}-e^{x}}{h}$
22. $\lim _{x \rightarrow 0}(1-x)^{\frac{1}{x}}$
23. $\lim _{x \rightarrow 0}(1+m x)^{\frac{m}{x}}$
24. $\lim _{x \rightarrow 0}\left(\frac{1+3 x}{1-5 x}\right)^{\frac{1}{x}}$
25. $\lim _{x \rightarrow 0} \frac{2^{x}-1}{\sqrt{1+x}-1}$
Type 2
Question 26
$\lim _{x \rightarrow 3} \frac{\log _{e}(x-2)}{x-3}$
Question 27
$\lim _{x \rightarrow 0} \frac{\log _{e}(1+2 x)}{x}$
Question 28
$\lim _{x \rightarrow 0} \frac{\log _{e}(1+x)}{3^{x}-1}$
Question 29
$\lim _{x \rightarrow 0} \frac{e^{\sin 3 x}-1}{\log (1+\tan 2 x)}$
Question 30
$\lim _{t \rightarrow 0} \frac{\log \left(1+t^{3}\right)}{\sin ^{3} t}$
Question 31
$\lim _{x \rightarrow 0} \frac{\sqrt{1+\sin 3 x}-1}{\log (1+\tan 2 x)}$
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