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Finding limits of a piecewise defined function

Calculus I Tutorial, by Dave Collins

I. From the graph

II. From the algebraic representation of the function

Let's start with the graph. Suppose you have the graph of a piecewise defined function:

f (x)

First, make sure you recall the algebra - being able to evaluate the function. For example,

can you determine the following?

f (-8) = f (2) =

f (-4) = f (4) =

f (-2) = f (6) =

f (0) = f (8) =

In this case, simply look at the graph and try to determine the exact value of y at the

desired x coordinate. Did you get the following solutions?

f (-8) = undefined f (2) = -3

f (-4) = 0 f (4) = 2

f (-2) = 2 f (6) = 5

f (0) = 0 f (8) = 7

Now, in Calculus we're concerned with values of y as x approaches a given value. For example,

lim f (x) = ?

x0

is asking, "What does the value of y start approaching as x gets close to zero?". To determine this,

just start tracing the graph (yes, you can use your fingers!) from both directions (the left and the

right) as x gets closer and closer to zero. The next series of graphs should give you the feel of this

tracing...

Since the y-value of the point seems to approach the same value from both sides, we say

that the limit does exist, and the value of this limit is y=0. You would write,

lim f (x) = 0

x0

It does not matter what happens at the point in order for the limit to exist!!!

Previously, it was shown that the value of the curve at x = 2 is y = -3. However, can you find the

limit of this function as x approaches 2? In other words, what is:

lim f (x) = ?

x2

Again, start tracing the function from both sides to see if the value of y approaches the

same number... The next series of graphs should depict what you should be visualizing:

So, in this example, even though we found the function value to be -3, you should be able to

visualize that it appears as if the value of y approaches 2, when x gets close to 2. Or,

lim f (x) = 2

x2

It must be emphasized that it does not matter the value of the function at 2 to

determine the limit. The next example shows that even though the value of the function exists, the

limit may not.

lim f (x) = ?

x6

Again, tracing the curve as x approaches 6 from both directions should show you that the

value of y does not appear to be the same:

So, the limit from the left appears to be y = 2, while the limit from the right appears to

be y = 8. Therefore, since these values are not the same, we say the limit does not exist.

lim f (x) = DNE (does not exist)

x6

This example does bring yet another important concept of limits - the existence of one

sided limits. You can restrict your attention to only approach a given value of x from one side -

either the left or the right. To denote this, to approach from the left use a superscript of a minus

sign, and to approach from the right use a superscript of a plus sign. For example, to ask, "What is

the limit of the function as x approaches 2 from the left?" you would see:

lim f (x) = ?

x6-

And from the work earlier, we know this limit is 2. Or,

lim f (x) = 2

x6-

How to write a piecewise function from a given graph? Remember A piecewise function is a function that is defined by different formulas or functions for each given interval. A non-void subset of A x B is a function from A to B if each element of A appears in some ordered pair in f and no two pairs ... Let f : A → B. ... The function f ( x) is defined by f ( x ) = | x | = {-x, x<0 x, x≥0 is called a modulus function. ... More items...

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