# Functions, Graphs, and Limits Examples

#### 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's cruising the web to find the perfect one. Eventually, she gives up...

#### 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, t...

#### Vertical Asymptotes vs. Holes

Both vertical asymptotes and holes are places that the curve can't quite seem to touch. Holes occur at places where the limit of the function exists, but the function itself does not. For rationa...

#### 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's a horizontal asympt...

#### 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 fu...

#### 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 f...

#### 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, bec...

#### 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)x1...

#### Multiplication by a Constant

If ,then .This rule is often written .We say that we're "pulling out'' the constant c from the limit.Sample ProblemFor example,and .Be Careful: This rule is only valid if is...

#### 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 both and exist, thenIn words, as long as the limits that are added...

#### 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.Here's a few more examples of this:Assume Th...

#### Powers and Roots of Limits

Limits are pretty powerful. They're kind of the big idea of calculus. Throughout calculus we'll see that no matter what we're doing, there's a limit or two lurking somewhere.The purpose of this re...