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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