# Calculus/Limits/Exercises

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

## Basic Limit Exercises

1. $\lim_{x\to 2} (4x^2 - 3x+1)$

$11$

2. $\lim_{x\to 5} (x^2)$

$25$

Solutions

## 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. $\lim_{x\to 7^-} |x^2+x| -x$

$49$

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

$0$

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

The limit does not exist

Solutions

## 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. $\lim_{x\to 0} \frac{\left|x\right|}{x}$

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. $\lim_{x\to 1} x^ + 5x - \frac{1}{2-x}$

$5$

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.

Solutions

## 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. $\lim_{x\to -\infty} 3x^2-2x+1$

$\infty$

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. $\lim_{x\to \infty} \frac{x^2+2}{x^3-2}$

$0$

Solutions

## Limits of Piecewise Functions

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

37. Consider the function

$f(x) = \begin{cases} (x-2)^2 & \mbox{if }x<2 \\ x-3 & \mbox{if }x\geq 2. \end{cases}$
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 & \mbox{if }x\leq 0 \\ x+1 & \mbox{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

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

$1$

Solutions