Last modified on 20 January 2014, at 04:53

Calculus/Limits/Exercises

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

Basic Limit ExercisesEdit

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

11

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

25

Solutions

One-Sided LimitsEdit

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 LimitsEdit

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 InfinityEdit

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 FunctionsEdit

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<x<4 \\ x^2 +2 & \mbox{if }x \geq 4. \end{cases}
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

External LinksEdit


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