Graphically+and+Numerically

Finding limits Graphically and Numerically

Finding Limits graphically and numerically can sometimes cause issues to beginning calculus students. This page is here to provide a simple and knowledgeable way to learn limits with continuous points and functions and with piecewise functions. xa || f(x) = L ||  || As x approaches the number a from the left, f(x) approaches the number L || xa || f(x) = R ||  || As x approaches the number a from the right, f(x) approaches the number R || If the left limit and the right limit //exist and are equal// (to L, say) then we say that **limx → af(x) exists and equals L**, and write xa || f(x) = L. ||  || As x approaches the number from both sides, f(x) approaches the single number L || Look at the following graph of the function f: From the graph, let us try to estimate limx→-2 f(x). If we were estimating the limit //numerically//, we would want a table that shows what is happening to the y-coordinates as the x-coordinates approach -2 from both sides. Since we are estimating the limit //graphicaly//, we instead use the following procedure.
 * Definition of a Limit**
 * lim
 * lim
 * lim
 * [[image:http://people.hofstra.edu/stefan_waner/realworld/tutorials/33bfig1.gif align="center"]] ||

|| || In the example illustrated, we are estimating limx→-2 f(x), so we have a = -2. Therefore, we position our trace point on the graph to the right of x = -2. || || || xa || f(x). || In the example we are doing, notice that the y-coordinate is approaching 2 as x approaches -2 from the right. (See the moving graphic on the right. If it has vanished, click on the graphic icon to reload it.) Therefore, x-2 || f(x) = 2. ||  ||  || || xa || f(x). || In the example we are doing, the y-coordinate is again approaching 2 as x approaches -2 from the left. Therefore, x-2 || f(x) = 2. ||  ||  || || xa || f(x) exists and equals L. || In our example, the left and right limits both exist and equal 2, and so
 * Estimating Limits Graphically** To decide whether limx→a f(x) exists, and to find its value if it does:
 * ** Step 1. ** Draw the graph of f(x) either by hand or using technology, such as a graphing calculator. || [[image:http://people.hofstra.edu/stefan_waner/realworld/tutorials/33bfig1.gif align="center"]] ||
 * ** Step 2. ** Position your pencil point (or the graphing calculator "trace" cursor) on a point of the graph to the right of x = a.
 * ** Step 3. ** Move the point //along the graph// toward x = a from the right and read the y-coordinate as you go. The value the y-coordinate approaches (if any) is then the limit || lim
 * lim
 * ** Step 4. ** Repeat Steps 2 and 3, but this time starting from a point on the graph to the left of x = a, and approach x = a along the graph from the left. The value the y-coordinate approaches (if any) is then || lim
 * lim
 * ** Step 5. ** If the left and right limits both exist and have the same value L, then || lim

xa || f(x) = 2. ||  || []
 * lim

Quick overview of what we've learned! media type="custom" key="11919720" =Piecewise Functions=

A Function Can be in Pieces
You can create functions that behave differently depending on the input (x) value.

A function made up of 3 pieces

Example: A function with three pieces:
It looks like this:
 * when x is less than 2, it gives **x2**,
 * when x is exactly 2 it gives **6**
 * when x is more than 2 and less than or equal to 6 it gives the line **10-x**

(a solid dot means "including",an open dot means "not including")

And this is how you write it:

The Domain is all Real Numbers up to and including 6: Dom(f) = (-∞, 6] (using [|Interval Notation])  Dom(f) = {x | x ≤ 6} (using [|Set Builder Notation])

And here are some example values: ||~ X
 * ~ Y ||
 * -4 || 16 ||
 * -2 || 4 ||
 * 0 || 0 ||
 * 1 || 1 ||
 * 2 || 6 ||
 * 3 || 7 ||

The Absolute Value Function
The [|Absolute Value Function] is a famous Piecewise Function. It has two pieces:
 * below zero: **-x**
 * from 0 onwards: **x**

f(x) = |x|

The Floor Function
The [|Floor Function] is a very special piecewise function. It has an infinite number of pieces:

The Floor Function

Use what you have learned to comprehend the below image.