- 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 the whole idea of limits has seemed a little too vague, then we're in luck. It's time to pull ourselves out of the calculus confusion. Let's don our prom attire and prepare for the formal, precise definition for 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.

This will make a lot more sense with a picture. Here's the idea: we're claiming that as *x* "gets closer" to *a*, *f*(*x*) "gets closer" to *L*, therefore the function would look something like this:

or perhaps like this, with a hole in the graph:

If we try to translate this formal definition into English, it's saying "no matter how close we want *f*(*x*) to be to *L*, if we pick *x* to be close enough to *a* we'll find the solution."

The real number ε is how close we want *f*(*x*) to be to *L*. That is, we want all the function values to fall within the shaded band:

The real number δ is how close *x* needs to be to *a* for the function values to fall within the shaded band:

Let *f*(*x*) = 2x. Then

If we take, as an example, ε = 0.5, how close does *x* need to be to *1* in order for *f*(*x*) = 2*x* to be within 0.5 of 2?

Saying we want *f*(*x*) to be within 0.5 of 2 means we want *f*(*x*) to be in between 1.5 and 2.5 (in the shaded band on the above graph).

For *f*(*x*) to be greater than 1.5 we need *x* to be greater than 0.75, and for *f*(*x*) to be less than 2.5 we need *x* to be less than 1.25. Drawing this on the graph, we see that if *x* is within 0.25 of 1 then*f*(*x*) will be in the shaded band.

If we took any other value of ε > 0, we could find some other value of δ > 0 so that |x-1| < δ would guarantee |*f*(*x*)-2| < ε. Therefore the limit of *f*(*x*) as *x* approaches 1 is equal to 2.

Let

This function looks like this:

It's tempting to say that

However, if we take a tiny value of ε, no matter how small we take δ to be, there will be some troublesome values of *f*(*x*). By troublesome we mean points where *x* is within δ of 1, but *f*(*x*) is *not* within ε of 5. Therefore the limit of *f*(*x*) as *x* approaches 1 can't possibly be 5.