- 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

Sometime measurements and results aren't perfect. This is true in science, engineering and life. How close to 16.9 fluid ounces of Mathlete-ade can a plastic bottle hold in order to allow 16.9 fluid ounces to be printed on the label? There is a certain range allowed by ShmoopCo that is sure to quench calculus induced thirst, but not too much to send a student to the little girls or boys room mid-class. Using limits, we can answer such questions.

One useful aspect of limits is something many calculus classes don't cover, or don't cover much: the **error**. To understand error, it helps to understand the formal definition of a limit.

**Definition.** Let *f*(*x*) be a function. We say the *limit of f(x) as x approaches a is L*, written

if for every real number ε > 0 there exists a real number δ > 0 such that

if |*x*-*a*| < δ then |*f*(*x*)-*L*| < δ.

We can think of ε as the error that's allowed in a measurement. If we know the measurement is the limit of some "nice'' function, the definition of limit says that we can choose the error ε that we want to allow and there will be some δ that will guarantee our measurement will have only the allowed amount of error.