Calculus/Limits/Exercises

Basic Limit Exercises

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

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

One-Sided Limits

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

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

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

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

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

Two-Sided Limits

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Limits to Infinity

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

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

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

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

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

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

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

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

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

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

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

Limits of Piecewise Functions

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

37. Consider the function

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

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

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

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

38. Consider the function

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

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

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

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

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

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

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

39. Consider the function

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

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

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

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

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

External Links

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