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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 la...

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'll shake things up a bit by taking limits with piecewise-defined functions. Here's an example: What is ?If we draw the graph of this function, we see that it looks like the line y = x + 1...

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...

Limits via Tables

There's 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 calculato...

Limits via Algebra

Most of the time, it's more precise (and a lot faster) to find limits using algebra. When finding a limit of the form  , where f(x) is just one nice algebraic expression, the first thing to...

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...

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...

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...
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