26 ασκήσεις ορίων με χρήση κανόνα De L΄ Hospital

1. $\underset{x\to 0}{lim}{\displaystyle \frac{\cos x-1}{\sin x}}$ 

2. $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{e^x}{x^3}}$

 

3. $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{\ln x}{x}}$ 

4. $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{\ln x}{x}}$ 

5. $\underset{x\to 0}{lim}{\displaystyle \frac{\sqrt{9+x}-3}{x}}$ 

6. $\underset{x\to 2}{lim}{\displaystyle \frac{2-\sqrt{x+2}}{4-x^2}}$ 

7. $\underset{x\to 1}{lim}{\displaystyle \frac{\sqrt{x}-1}{\sqrt[3]{x}-1}}$ 

8. $\underset{x\to 0}{lim}{\displaystyle \frac{x^2}{\sqrt{2x+1}-1}}$ 

9. $\underset{u\to 1}{lim}{\displaystyle \frac{(u-1{)}^3}{(1/u)-u^2+(3/u)-3}}$ 

10. $\underset{x\to 0}{lim}{\displaystyle \frac{2+(1/x)}{3-(2/x)}}$ 

11. $\underset{x\to 0^{+}}{lim}{\displaystyle \frac{1+5/\sqrt{x}}{2+1/\sqrt{x}}}$ 

12. $\underset{x\to \pi /2}{lim}{\displaystyle \frac{\cos x}{(\pi /2)-x}}$ 

13. $\underset{x\to 0}{lim}{\displaystyle \frac{x^2}{e^x-x-1}}$ 

14. $\underset{x\to 1}{lim}{\displaystyle \frac{\ln x}{x-1}}$ 

15. $\underset{x\to 0}{lim}{\displaystyle \frac{\ln (x^2+1)}{x}}$ 

16. $\underset{x\to 1}{lim}{\displaystyle \frac{x\ln x}{x^2-1}}$ 

17. $\underset{x\to 0}{lim}{\displaystyle \frac{\sin (2x)}{\ln (x+1)}}$ 

18. $\underset{x\to 1}{lim}{\displaystyle \frac{\sqrt{x}-1}{x-1}}$ 

19. $\underset{x\to 0}{lim}{\displaystyle \frac{\sqrt{x+1}-1}{\sqrt{x+4}-2}}$ 

20. $\underset{x\to 0}{lim}{\displaystyle \frac{\sqrt{x^2+1}-1}{\sqrt{x+1}-1}}$ 

21. $\underset{x\to 0^{+}}{lim}\sqrt{x}\ln x$

22. $\underset{x\to 0}{lim}{\displaystyle \frac{(1-x{)}^{1/4}-1}{x}}$

23. $\underset{t\to 0}{lim}(t+{\displaystyle \frac{1}{t}})((4-t{)}^{3/2}-8)$

24. $\underset{t\to 0^{+}}{lim}({\displaystyle \frac{1}{t}}+\frac{1}{\sqrt{t}})(\sqrt{t+1}-1)$

25. $\underset{x\to 0}{lim}{\displaystyle \frac{e^x-1}{x}}$

26. $\underset{x\to 1}{lim}{\displaystyle \frac{x^{1/4}-1}{x}}$