- 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

Now we will shake things up a bit. Here's a piecewise-defined function:

What is ?

If we draw the graph of this function, we see that it looks like the line *y* = *x* + 1 except at one point. When *x* = 1, instead of having *y* = 2 like we would expect, the point has jumped off the line up to *y* = 3.

How does a function like this affect what we know about limits? Imagine we're taking Bruno, the Chinese crested dog, for a walk. We would expect him to stay on the sidewalk. We wouldn't expect him to suddenly teleport to Middle-earth, then reappear and continue on his path. He may *look* like Gollum, but still...When talking about limits, we're talking about what we expect the function to be doing. We assume Bruno is approaching solid ground.

In the example above, because that's what we would expect the value of the function to be if we looked at values of *x* close to (but not equal to) 1.

We can think of as the value that *f(x)* gets "close" to as *x* gets close to 1.

Exercise 1

For the function, find the specified limit.

*lim* _{x to 1 }*f*(*x*)

Exercise 2

For the function, find the specified limit.

*lim* _{x to 1 }*g*(*x*)

Exercise 3

For the function, find the specified limit.

*lim* _{x to 1} *h*(*x*)