NAME _________________________________________ DATE ______________ PERIOD _______

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.**

*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.**;***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.**

- ;
*x*= 2 *f*(*x*) =*x*^{2}+ 5*x*+ 3;*x*= 4

Chapter 1 | 16 |
Glencoe Precalculus |

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) = x^{3} + 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 |