- 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

Who wins when we compare polynomials and logarithmic functions? Look at a picture.

Eventually, after not too long, the polynomial will pull ahead of the logarithmic function. This makes sense, because the polynomial is curved upwards, while the logarithmic curve looks like it's flattening out.

The logarithmic curve never flattens out, it has no horizontal asymptotes, and grows without bound, but it does so slowly.

If we take some limits, we find

and

Think of these three types of functions as if they are racing. Power functions are like powerful race horses; polynomials (Polly want a cracker?) are like parrots fluttering along; and logarithmic functions are like logs, plodding and slow. The category a function belongs to is determined by its *leading term*, that is, the individual term that "grows the fastest'."

Example 1

Find |

Exercise 1

Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.

- 4
*x*^{2}+ 5*x*^{7}+ 3^{x}

Exercise 2

Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.

- ln
*x*+*x*

Exercise 3

Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.

*x*^{2}+*x*^{3}

Exercise 4

- ln
*x*+*x*^{99}+ 9^{x}

Exercise 5

- -4
^{x}+ 2^{x}

Exercise 6

Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.

Exercise 7

Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.

Exercise 8

Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.

Exercise 9

Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.

Exercise 10

Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.