- Topics At a Glance
- Limits
- Functions Are Your Friend
- Graphing and Visualizing Limits
- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
- All About Asymptotes
- Vertical Asymptotes
- Finding Vertical Asymptotes
- Vertical Asymptotes vs. Holes
- Limits at Infinity
- Natural Numbers
- Limits Approaching Zero
- Estimating a Circle
- The Cantor Set and Fractals
- Limits of Functions at Infinity
- Horizontal, Slant, and Curvilinear Asymptotes
- Horizontal Asymptoes
- Slant Asymptotes
- Curvilinear Asymptotes
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Comparing Functions
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
**Manipulating Limits****The Basic Properties**- Multiplication by a Constant
- Adding and Subtracting Limits
- Multiplying and Dividing Limits
- Powers and Roots of Limits
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

If *c* is a real number, then

Think of this as taking the limit of the constant function *f*(*x*) = c. No matter what we plug in for *x*, we get *c* as the output. If we made a table

x | f(x) |
---|---|

x1 | c |

x2 | c |

every number in the *f*(*x*) column would be *c*. As *x* gets closer to *a*, *f*(*x*) gets closer to (or stays equal to) *c*.

As long as *a* is a real number, it doesn't even matter what *a* is. *c* is all that matters.

Using actual numbers, what is ?

We're looking at the limit as *x* approaches 3 of the function *f*(*x*) = x. If we make a table, we find

x | f(x) |
---|---|

2.99 | 2.99 |

2.999 | 2.999 |

3.0001 | 3.0001 |

3.001 | 3.001 |

3.01 | 3.01 |

As *x* gets closer to 3, *f*(*x*) gets closer to 3 since *x* and *f*(*x*) are the same thing.

The picture of *f*(*x*) = *x* is a line:

From the picture we can see that as *x* gets closer to 3, *f*(*x*) also gets closer to 3.

We conclude that = 3.

Example 1

Find \lim_x\to 5x. |

Exercise 1

Find the limit.

\lim_x\to -30 π

Exercise 2

\lim_x\to 0\sqrt2

Exercise 3

\lim_x\to 10x

Exercise 4

\lim_x\to -4x

Exercise 5

\lim_x\to 10

Exercise 6

Find the following limit.

\lim_x\to2x

Exercise 7

\lim_x\to23x

Exercise 8

\lim_x\to210x

Exercise 9

\lim_x\to2x^{2} + 5

Exercise 10

\lim_x\to23x^{2} + 15

Exercise 11

\lim_x\to210x^{2} + 50