- 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

There is more than one way to approach (pun absolutely intended) limit problems. We've already looked at graphs and equations.

Another way to estimate the limit of a function is to use a calculator to see what the function approaches as we plug in values of *x* that get closer and closer to some value *a*. To keep things organized, now we'll use tables to get the lowdown on functions.

If *f*(*x*) = *x*^{2}, estimate

.

We make a table. In one column we'll have values of *x*, and in the next we'll have the corresponding values of *f*.

First we have *x* approach 3 from the left.

The values of *f*(*x*) in the table appear to be getting closer to 9 as *x* approaches 3 from the left. We'll see what happens if *x* approaches 3 from the right.

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

2.5 | 6.25 |

2.7 | 7.29 |

2.9 | 7.29 |

2.9 | 8.41 |

2.99 | 8.9401 |

2.999 | 8.994001 |

The values of *f*(*x*) appear to be approaching 9 as *x* approaches 3 from the right, also. We can now shout from the rooftops that, indeed, = 9.

When using tables to determine limits, there's no particular rule about what numbers to plug in for *x* as it approaches a number *a*. As long as we look at lots of values of *x*, and let them get really close (as in, 0.00001 close) to *a*, we should be fine.

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

3.1 | 9.61 |

3.01 | 9.0601 |

3.001 | 9.006001 |

3.0001 | 9.0006001 |

Exercise 1

Let *f*(*x*) = \frac*x*^{3} + 2*x*^{2}-5*x*-6*x*^{3} + 3*x*^{2}-6*x*-8 Use numerical tables to estimate the limit.

- \lim_x\to -1
*f*(*x*)

Exercise 2

Let *f*(*x*) = \fracx^{3} + 2x^{2}-5x-6x^{3} + 3x^{2}-6x-8 Use numerical tables to estimate the limit.

- \lim_x\to 1
*f*(*x*)

Exercise 3

Let *f*(*x*) = \fracx^{3} + 2x^{2}-5x-6x^{3} + 3x^{2}-6x-8 Use numerical tables to estimate the limit.

\item \lim_x\to 2*f*(*x*)

Exercise 4

Let *f*(*x*) = \fracx^{3} + 2x^{2}-5x-6x^{3} + 3x^{2}-6x-8 Use numerical tables to estimate the limit.

\item \lim_x\to -3*f*(*x*)