- 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

Functions are machines. Plug the independent variable into the machine and it spits out the dependent variable.

If *y* = *f*(*x*) = *x* + 1, then as *x* gets larger (moves right), *y* gets larger also (moves up). As *x* gets smaller (moves left), *y* gets smaller also (moves down).

Here's something to play with: see what happens to *y* as we make *x* larger or smaller.

If *y* = *f*(*x*) = 1 - *x*, then as *x* gets larger (moves right), *y* gets smaller (moves down). As *x* gets smaller (moves right), *y* gets larger (moves up).

Say *y* = *f*(*x*) = *x*^{2}. We start *x* at -5. As *x* moves right, *y* gets smaller until *x* reaches 0. If, starting at 0, we keep moving *x* to the right, *y* starts getting bigger again.

With this function, in order to say whether *y* is increasing or decreasing as we play with *x*, we need to know two things: whether *x* is to the left or the right of zero, and whether *x* is being moved to the right or left.

Exercise 1

Let *y* = *f*(*x*) = -*x*^{2} + 1.

- If
*x*is greater than zero, and getting larger, is*y*getting larger or smaller?

Exercise 2

Let *y* = *f*(*x*) = -*x*^{2} + 1.

- If
*x*is greater than zero, and getting smaller, is*y*getting larger or smaller?

Exercise 3

Let *y* = *f*(*x*) = -*x*^{2} + 1.

- If
*x*is less than zero, and getting smaller, is*y*getting larger or smaller?

Exercise 4

Let *y* = *f*(*x*) = -*x*^{2} + 1.

- If
*x*is less than zero, and getting larger, is*y*getting larger or smaller?

Exercise 5

Let *y* = *f*(*x*) = -*x*^{2} + 1.

- As
*x*gets close to zero, what does*y*approach?