Basic Limit Exercises

1. ${\displaystyle \lim _{x\to 2}{\Big [}4x^{2}-3x-1{\Big ]}}$

2. ${\displaystyle \lim _{x\to 5}{\Big [}x^{2}{\Big ]}}$

One-Sided Limits

Evaluate the following limits or state that the limit does not exist.

3. ${\displaystyle \lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}}$

4. ${\displaystyle \lim _{x\to 7^{-}}{\Big [}|x^{2}+x|-x{\Big ]}}$

5. ${\displaystyle \lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}}$

6. ${\displaystyle \lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}}$

Two-Sided Limits

Evaluate the following limits or state that the limit does not exist.

7. ${\displaystyle \lim _{x\to -1}{\frac {1}{x-1}}}$

8. ${\displaystyle \lim _{x\to 4}{\frac {1}{x-4}}}$

9. ${\displaystyle \lim _{x\to 2}{\frac {1}{x-2}}}$

10. ${\displaystyle \lim _{x\to -3}{\frac {x^{2}-9}{x+3}}}$

11. ${\displaystyle \lim _{x\to 3}{\frac {x^{2}-9}{x-3}}}$

12. ${\displaystyle \lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}}$

13. ${\displaystyle \lim _{x\to -1}{\frac {x^{3}+1}{x+1}}}$

14. ${\displaystyle \lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}}$

15. ${\displaystyle \lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}}$

16. ${\displaystyle \lim _{x\to 0}{\frac {|x|}{x}}}$

17. ${\displaystyle \lim _{x\to 2}{\frac {1}{(x-2)^{2}}}}$

18. ${\displaystyle \lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}}$

19. ${\displaystyle \lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}}$

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

21. ${\displaystyle \lim _{x\to 3}{\frac {x+3}{x^{2}-9}}}$

22. ${\displaystyle \lim _{x\to -1}{\frac {x+1}{x^{2}+x}}}$

23. ${\displaystyle \lim _{x\to 1}{\frac {1}{x^{2}+1}}}$

24. ${\displaystyle \lim _{x\to 1}\left[x^{2}+5x-{\frac {1}{2-x}}\right]}$

25. ${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}}$

26. ${\displaystyle \lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}}$

Limits to Infinity

Evaluate the following limits or state that the limit does not exist.

27. ${\displaystyle \lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}}$

28. ${\displaystyle \lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}}$

29. ${\displaystyle \lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}}$

30. ${\displaystyle \lim _{x\to -\infty }{\Big [}3x^{2}-2x+1{\Big ]}}$

31. ${\displaystyle \lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}}$

32. ${\displaystyle \lim _{x\to \infty }6}$

33. ${\displaystyle \lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}}$

34. ${\displaystyle \lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}}$

35. ${\displaystyle \lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}}$

36. ${\displaystyle \lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}}$

Limits of Piecewise Functions

Evaluate the following limits or state that the limit does not exist.

37. Consider the function

${\displaystyle f(x)={\begin{cases}(x-2)^{2}&{\text{if }}x<2\\x-3&{\text{if }}x\geq 2\end{cases}}}$

a. ${\displaystyle \lim _{x\to 2^{-}}f(x)}$

b. ${\displaystyle \lim _{x\to 2^{+}}f(x)}$

c. ${\displaystyle \lim _{x\to 2}f(x)}$

38. Consider the function

${\displaystyle g(x)={\begin{cases}-2x+1&{\text{if }}x\leq 0\\x+1&{\text{if }}0<x<4\\x^{2}+2&{\text{if }}x\geq 4\end{cases}}}$

a. ${\displaystyle \lim _{x\to 4^{+}}g(x)}$

b. ${\displaystyle \lim _{x\to 4^{-}}g(x)}$

c. ${\displaystyle \lim _{x\to 0^{+}}g(x)}$

d. ${\displaystyle \lim _{x\to 0^{-}}g(x)}$

e. ${\displaystyle \lim _{x\to 0}g(x)}$

f. ${\displaystyle \lim _{x\to 1}g(x)}$

39. Consider the function

${\displaystyle h(x)={\begin{cases}2x-3&{\text{if }}x<2\\8&{\text{if }}x=2\\-x+3&{\text{if }}x>2\end{cases}}}$

a. ${\displaystyle \lim _{x\to 0}h(x)}$

b. ${\displaystyle \lim _{x\to 2^{-}}h(x)}$

c. ${\displaystyle \lim _{x\to 2^{+}}h(x)}$

d. ${\displaystyle \lim _{x\to 2}h(x)}$

External Links

• Limits 1 exercise at Khan Academy
• Limits 2 exercise at Khan Academy