- Functions Are Your Friend
- Graphing and Visualizing Limits
- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
- Vertical Asymptotes
- Finding Vertical Asymptotes
- Vertical Asymptotes vs. Holes
- Limits of Functions at Infinity
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
- The Basic Properties
- Adding and Subtracting Limits
- Multiplying and Dividing Limits
- Powers and Roots of Limits
- Best of the Web
Functions Are Your Friend
Functions are machines. Plug the independent variable into the machine and it spits out the dependent variable.Sample ProblemIf y = f(x) = x + 1, then as x gets larger (moves right), y gets larger...
Graphing and Visualizing Limits
It's super helpful to plug numbers into the function and see the output. It's even more helpful to graph the results. Try to draw or imagine how a function actually looks. Is it really hip...
Piecewise Functions and Limits
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 =...
Becky has been planning her Florida vacay for months. The only thing left on her to-do list is to find a new bathing suit. She is cruising the web to find the perfect one. Eventually, she gives up...
Limits via Tables
There is more than one way to approach (pun absolutely intended) limit problems. We've already looked at graphs and equations. Another way to estimate the limit of a function is to use a calculator...
Limits via Algebra
Most of the time, it's more precise to find the limit using algebra. When finding a limit of the form , the first thing to do is plug a into the function to see if it exists. If f(a) exists,...
Now we'll check out one of the rock stars of the limit world:To the right, to the right, to the right, to the rightWe'll look at this limit one side at a time. First we'll look at the limit as x ap...
Finding Vertical Asymptotes
Vertical asymptotes most frequently show up in rational functions. When a rational function f(x) has a non-zero constant in the numerator and an expression with a variable in the denominator, the f...
Vertical Asymptotes vs. Holes
Both vertical asymptotes and holes are places where the curve never quite touches. Holes occur at places where the limit of the function exists but the function itself does not. These correspond to...
Limits of Functions at Infinity
When we find the limit of a function f(x) as x goes to infinity, we're answering the question "What value is f(x) approaching as x gets bigger and bigger and bigger...?''A rockin' example...
Finding Horizontal/ Slant/ Curvilinear Asymptotes
Sometimes when a function has a horizontal asymptote, we can see what it should be. Sample ProblemLet f(x) = 4-x. Then as x approaches ∞ the function f approaches 0, there is a horizontal asympto...
How to Draw Rational Functions from Scratch
We now have enough tools to draw some complicated functions from scratch. Now we know how graphing calculators do it, and why they require the energy of four triple-A's.When drawing a rational func...
Power Functions vs. Polynomials
In the world of horse races, power functions like 2x will always grow faster than plain old polynomials, no matter how high the degree of the polynomial. By "grow faster'' we mean that if we go far...
Polynomials vs. Logarithmic Functions
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, becau...
The Basic Properties
Basic Property 1If c is a real number, then Think of this as taking the limit of the constant function f(x) = c. No matter what we plug in for x, we get c as the output. If we made a tablexf(x)x1cx...
Adding and Subtracting Limits
Limits can be added and subtracted, but only when those limits exist.Adding and Subtracting Property 1If a is a real number and bothand exist, thenIn words, as long as the limits that are added bot...
Multiplying and Dividing Limits
Multiplication PropertyAs long as both and exist,In words, the limit of a product is the product of the limits, as long as the limits involved exist.Assume Then Assume ThenAssume Then Using...
Powers and Roots of Limits
Power PropertyIf exists, and p is any real number,Sample ProblemIf then