Finding limits of a piecewise defined function Calculus I … - limit of piecewise function calculator

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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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