AP Calculus (AB)
A LIMIT IS
…
A limit is a number (or an algebraic
expression ... see below).
On a graph, a limit is a y-value.
It is a particular y-value.
It is the specific y-value associated
with a particular x-value.
It is the y-value that the function
outputs are getting close to as we creep closer and closer to
a particular x-value.
In the graph shown here, it appears as x gets
closer and closer to 1 that the y-values are getting closer
and closer to 5. Thus, we can make a conjecture that the limit
of the function as x approaches 1 is 5.
The number along the x-axis that we
are approaching is generally called c. If we can find
a limit as we approach c, the limit is called L.
The formal mathematical notation to say, "the
limit of a function, f, as x approaches some number
c is the number L," is:
We can discuss "one-sided limits,"
such as the limit as we approach c only from the left
or only from the right. The notations for these are
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Limit from the left >>> |
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Limit from the right >>> |
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For a limit to exist, the limit from the left
must equal the limit from the right.
This graph does
have a limit as x approaches p/2.
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This graph does not
have a limit as x approaches p/2.
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When the limit from the left and the limit
from the right equal different values, the limit does not exist
and we call the "break" in the graph a "gap" or
a jump discontinuity.
It does not matter whether the function has
a y-value at the place where we say there is a limit.
There could be a hole in the graph! The y-value of the location
of the hole is the limit.
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Functions that have limits at "holes"
in their graph. |
the limit exists at x = 2 |
the limit exists at x = 2 |
the limit exists at x = 3 |
These "holes" in the graphs could
be filled in by one point, such points are called "removable points of discontinuity."
Limits at Endpoints:
Functions that have restricted domains, with
endpoints, will have limits only from the left or from the right
at endpoints. For example, a radical function will be defined
only when the radicand is non-negative. Many radical functions
will have restricted domains with endpoints. Since it does not
make sense to speak of the limits of the graph on intervals where
the function does not exist, when speaking of a limit at an endpoint,
if the limit exists from the right or left as x approaches
the point, the limit at the endpoint will exist. It is not necessary
to hold the endpoint limit to the test of a "limit from
the left and a limit from the right" since the function
only exists on one side of the endpoint.
Functions that have limits at
endpoints of their graphs. |
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At the endpoint (–2, 0), the function only
exists to the right of the point, thus since the function approaches
0 as x approaches –2 from the right, the limit is
0.
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At the endpoint (3, 0), the function only exists
to the left of the point, thus since the function approaches
0 as x approaches 3 from the left, the limit is 0.
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If x approaches some value c
and the output values of the function steadily grow to either
positive or negative infinity, then the function is said to "increase
(or decrease) without bound." When this occurs, the graph
as a vertical asymptote and the limit fails to exist for that
value c.
"Gaps" (jump discontinuities) and
places where there exist vertical asymptotes in a graph are called
"nonremovable discontinuities" because the graph cannot be made continuous
by filling in the discontinuity with a single point.
Often, as we approach c, the y-values
do not approach one specific value. Some functions oscillate
wildly between different values when x approaches certain
values. When this happens, the limit does not exist even though
the function's graph is smooth and continuous.
The graphs below, of y = sin(1/x),
demonstrate the lack of a limit (as x approaches 0) due
to oscillating behavior.
Be careful when saying a function does or
does not have a limit. Limits are always associated with
some particular x = c. To say the function f
has a limit is meaningless ... until you say what
x is approaching.
Sometimes
a Limit is an Algebraic Expression
When the expression for which we are finding
the limit includes "other variables" (one is approaching
something and another that is in the expression), the results
will be an algebraic expression. For example, we shall soon learn
that the following limit is true.
The proof of this is somewhat tricky, involving
some other limits and some trigonometric identities, but it is
an extremely important limit. So, not all limits are numbers.
The mathematical concept
of a limit is the engine that allows calculus to work and is
one of the great achievements of human thought and science!
The formal definition of limits is called
the "epsilon-delta" definition. For more on those,
the following hyperlinks are recommended:
This page was created on May 19, 2002.
Last updated: March 18, 2004.
