# Calculus/Limits/Exercises

 ← Proofs of Some Basic Limit Rules Calculus Differentiation → Limits/Exercises

## Basic Limit Exercises

1. $\lim _{x\to 2}{\Big [}4x^{2}-3x+1{\Big ]}$ $11$ 2. $\lim _{x\to 5}{\Big [}x^{2}{\Big ]}$ $25$ ## One-Sided Limits

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

3. $\lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}$ ${\frac {1}{2}}$ 4. $\lim _{x\to 7^{-}}{\Big [}|x^{2}+x|-x{\Big ]}$ $49$ 5. $\lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}$ $0$ 6. $\lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}$ The limit does not exist

## Two-Sided Limits

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

7. $\lim _{x\to -1}{\frac {1}{x-1}}$ $-{\frac {1}{2}}$ 8. $\lim _{x\to 4}{\frac {1}{x-4}}$ The limit does not exist.

9. $\lim _{x\to 2}{\frac {1}{x-2}}$ The limit does not exist.

10. $\lim _{x\to -3}{\frac {x^{2}-9}{x+3}}$ $-6$ 11. $\lim _{x\to 3}{\frac {x^{2}-9}{x-3}}$ $6$ 12. $\lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}$ $0$ 13. $\lim _{x\to -1}{\frac {x^{3}+1}{x+1}}$ $3$ 14. $\lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}$ ${\frac {13}{8}}$ 15. $\lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}$ $10$ 16. $\lim _{x\to 0}{\frac {|x|}{x}}$ The limit does not exist.

17. $\lim _{x\to 2}{\frac {1}{(x-2)^{2}}}$ $\infty$ 18. $\lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}$ The limit does not exist.

19. $\lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}$ $-{\frac {5}{2}}$ 20. $\lim _{x\to 2}{\frac {x^{2}+2x+1}{x^{2}-2x+1}}$ $9$ 21. $\lim _{x\to 3}{\frac {x+3}{x^{2}-9}}$ The limit does not exist.

22. $\lim _{x\to -1}{\frac {x+1}{x^{2}+x}}$ $-1$ 23. $\lim _{x\to 1}{\frac {1}{x^{2}+1}}$ ${\frac {1}{2}}$ 24. $\lim _{x\to 1}\left[x^{2}+5x-{\frac {1}{2-x}}\right]$ $5$ 25. $\lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}$ ${\frac {1}{2}}$ 26. $\lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}$ The limit does not exist.

## Limits to Infinity

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

27. $\lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}$ $0$ 28. $\lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}$ ${\frac {1}{3}}$ 29. $\lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}$ ${\frac {3}{2}}$ 30. $\lim _{x\to -\infty }{\Big [}3x^{2}-2x+1{\Big ]}$ $\infty$ 31. $\lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}$ $0$ 32. $\lim _{x\to \infty }6$ $6$ 33. $\lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}$ $0$ 34. $\lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}$ ${\frac {3}{2}}$ 35. $\lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}$ $-\infty$ 36. $\lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}$ $0$ ## 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. $\lim _{x\to 2^{-}}f(x)$ $0$ b. $\lim _{x\to 2^{+}}f(x)$ $-1$ c. $\lim _{x\to 2}f(x)$ The limit does not exist

38. Consider the function

$g(x)={\begin{cases}-2x+1&{\text{if }}x\leq 0\\x+1&{\text{if }}0 a. $\lim _{x\to 4^{+}}g(x)$ $18$ b. $\lim _{x\to 4^{-}}g(x)$ $5$ c. $\lim _{x\to 0^{+}}g(x)$ $1$ d. $\lim _{x\to 0^{-}}g(x)$ $1$ e. $\lim _{x\to 0}g(x)$ $1$ f. $\lim _{x\to 1}g(x)$ $2$ 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. $\lim _{x\to 0}h(x)$ $-3$ b. $\lim _{x\to 2^{-}}h(x)$ $1$ c. $\lim _{x\to 2^{+}}h(x)$ $1$ d. $\lim _{x\to 2}h(x)$ $1$ 