Introduction to limits continuity and end behavior
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Introduction to Limits, Continuity , and End Behavior. Honors Precalculus Functions Mr. Frank Sgroi , M.A.T. Holy Cross high School.

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Introduction to limits continuity and end behavior

Introduction to Limits, Continuity , and End Behavior

Honors Precalculus Functions

Mr. Frank Sgroi, M.A.T.

Holy Cross high School


Intuitive definition of a limit

The limit of a function (if it exists) is a number that the y-value of a function approaches as the x-value approaches a specific number. The diagram below may be of assistance.

Notice: As x gets closer to 2 (from either direction), y gets closer to 2 (from both directions). So the limit =2.

Intuitive Definition of a Limit


Limit notation

We express this concept in symbolic notation as follows:

Read: The limit of f(x) as x approaches c is L

From the previous example the notation would be:

Limit Notation


The concept of continuity

A function is said to be continuous if “you can draw the graph of a function without lifting your pencil off of the paper”. A few examples are:

The Concept of Continuity


Discontinuous functions

A function that is NOT continuous is said to be discontinuous. A few examples are:

Jump

Discontinuity

Infinite

Discontinuity

Removable

Discontinuity

Discontinuous Functions


Continuity

Knowing whether a function is continuous or discontinuous is very useful when analyzing the graph and properties of a function.

To completely understand the idea of a continuous function we must construct a formal definition.

Continuity


Definition of continuity

A function very useful when analyzing the graph and properties of a function.f is continuous at x = c if and only if each of the following conditions is met:

(i) f(x) is defined at x = c.

If any of the above conditions is not met then the function is said to be discontinuous at x = c.

Definition of Continuity


Examples of discontinuity

The following functions are NOT continuous because… very useful when analyzing the graph and properties of a function.

The function is NOT defined at x = 2.

Examples of Discontinuity


End behavior of a function

One of the applications of limits is to describe the “end behavior” of a function. This idea explains how functions behave as the values of x “increase without bound” or “decrease without bound”.

End Behavior of a Function


End behavior example

Consider the graph of the function behavior” of a function. This idea explains how functions behave as the values of x “increase without bound” or “decrease without bound”.

As x moves to the right the values of y become larger. Using limits…

As x moves to the left the values of y become larger.

Using limits …

End Behavior (Example)


End behavior example1

Consider the graph of the function shown. behavior” of a function. This idea explains how functions behave as the values of x “increase without bound” or “decrease without bound”.

The end behavior for this function is described as follows:

End Behavior Example