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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.

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Introduction to Limits, Continuity , and End Behavior

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

  2. 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

  3. 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

  4. 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

  5. A function that is NOT continuous is said to be discontinuous. A few examples are: Jump Discontinuity Infinite Discontinuity Removable Discontinuity Discontinuous Functions

  6. 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

  7. 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

  8. The following functions are NOT continuous because… The function is NOT defined at x = 2. Examples of Discontinuity

  9. 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

  10. Consider the graph of the function 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)

  11. Consider the graph of the function shown. The end behavior for this function is described as follows: End Behavior Example

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