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Continuity, End Behavior, and Limits

Continuity

A function f(x) is continuous at x = c if it satisfies the following conditions.

(1) f(x) is defined at c; in other words, f(c) exists.

(2) f(x) approaches the same function value to the left and right of c; in other words, exists.
(3) The function value that f(x) approaches from each side of c is f(c); in other words, .

Functions that are not continuous are discontinuous. Graphs that are discontinuous can exhibit infinite discontinuity, jump discontinuity, or removable discontinuity (also called point discontinuity).

Example Determine whether each function is continuous at the given x-value. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

  1. f(x) = 2│x│ + 3; x = 2

    (1) f(2) = 7, so f(2) exists.
    (2) Construct a table that shows values for f(x) for x-values approaching 2 from the left and from
          the right.

    x y = f(x)
        1.9     6.8
        1.99     6.98
        1.999     6.998
    x y = f(x)
        2.1     7.2
        2.01     7.02
        2.001     7.002

          The tables show that y approaches 7 as x approaches 2 from both sides.

          It appears that .
    (3) and f(2) = 7.
          The function is continuous at x = 2.


  2. ; x = 1

    The function is not defined at x = 1 because it results in a denominator of 0. The tables show that for values of x approaching 1 from the left, f(x) becomes increasingly more negative. For values approaching 1 from the right, f(x) becomes increasingly more positive.



    x y = f(x)
    0.9 -9.5
    0.99 -99.5
    0.999 -999.5
    x y = f(x)
    1.1 10.5
    1.01 100.5
    1.001 1000.5

    The function has infinite discontinuity at x = 1.

Exercises

Determine whether each function is continuous at the given x-value. Justify your answer using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

  1. ; x = 2



  2. f(x) = x2 + 5x + 3; x = 4


Chapter 1 16 Glencoe Precalculus


End Behavior

The end behavior of a function describes how the function behaves at either end of the graph, or what happens to the value of f(x) as x increases or decreases without bound. You can use the concept of a limit to describe end behavior.

Left-End Behavior (as x becomes more and more negative):
Right-End Behavior (as x becomes more and more positive):
The f(x) values may approach negative infinity, positive infinity, or a specific value.

Example Use the graph of f(x) = x3 + 2 to describe its end behavior. Support the conjecture numerically.

As x decreases without bound, the y-values also decrease without bound. It appears the limit is negative infinity: .

As x increases without bound, the y-values increase without bound. It appears the limit is positive infinity: .


Construct a table of values to investigate function values as │x│ increases.

x -1000 -100 -10 0 10 100 1000
f(x) -999,999,998 -999,998 -998 2 1002 1,000,002 1,000,000,002

As x → -∞, f(x) → -∞. As x → ∞, f(x) → ∞. This supports the conjecture.

Exercises

Use the graph of each function to describe its end behavior. Support the conjecture numerically.








Chapter 1 17 Glencoe Precalculus