Calculus/Limits/Exercises

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

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

3. ${\displaystyle \lim _{x\to {\frac {\pi }{4}}}{\Big [}\cos ^{2}(x){\Big ]}}$

4. ${\displaystyle \lim _{x\to 1}{\Big [}5e^{x-1}-5{\Big ]}}$

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

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

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

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

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

9. ${\displaystyle \lim _{x\to 0^{-}}{\frac {|x|}{x}}}$

10. ${\displaystyle \lim _{x\to 1^{-}}\arcsin(x)}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

32. ${\displaystyle \lim _{x\to -1}\ln({\sqrt {x^{2}-1}})}$

33. ${\displaystyle \lim _{x\to 1}\arcsin(x)}$

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

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

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

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

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

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

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

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

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

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

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

44. ${\displaystyle \lim _{x\to \infty }{\frac {\sin \left({\frac {1}{x}}\right)}{x}}}$

45. ${\displaystyle \lim _{x\to -\infty }x^{2}\cos \left({\frac {1}{x}}\right)}$

46. ${\displaystyle \lim _{x\to \infty }{\frac {\sin(\arctan(x))}{\arctan(-x)}}}$

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

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

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

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

51. Use the intermediate value theorem to show that there exists a value ${\displaystyle f(x)=3}$ for ${\displaystyle f(x)={\sqrt {x}}+{\frac {1}{\sqrt {x}}}}$ from ${\displaystyle 1\leq x\leq 9}$. If you cannot use the intermediate value theorem to show this, explain why.

52. Use the intermediate value theorem to show that there exists an ${\displaystyle x=\mu }$ so that ${\displaystyle f(\mu )={\frac {\pi }{4}}}$ for ${\displaystyle f(x)={\frac {\sin(x)-\cos(x)}{\ln {x}}}}$ from ${\displaystyle \pi \leq x\leq {\frac {9}{4}}\pi }$. If you cannot use the intermediate value theorem to show this, explain why.

53. Use the intermediate value theorem to show that there exists a value ${\displaystyle x=\Gamma }$ so that ${\displaystyle f(\Gamma )=2}$ for ${\displaystyle f(x)={\sqrt {-2x+4}}+5\ln \left(x^{2}\right)}$ from ${\displaystyle -1\leq x\leq 1}$. If you cannot use the intermediate value theorem to show this, explain why.

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