# Functions, Graphs, and Limits

# Functions, Graphs, and Limits Exercises

#### 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 to be x2...

#### 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 =...

#### One-Sided Limits

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,...

#### Vertical Asymptotes

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 of this i...

#### 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