Unit 5 Limit Analysis

Given a function f(x), a limit expresses the value of f(x) as x approaches a certain value. A limit is expressed with the following format for a function f(x): (see right)

• α can be any value from -∞ to ∞ included
• f(x) is the function
• β is value of the limit

Verbally, a limit can be expressed as follows: f(x) as x approaches a.

A limit that describes a graph's end behavior describes the graph of a function as the inputs of f(x) reach positive and negative infinity.

For example, for the limit (see right), as x approaches ∞, or x→∞, or x becomes unbounded positively, the denominator (x-1) increases infinitely (becomes unbounded positively) as well. -50 divided by an increasingly large number results in a fraction getting infinitely smaller, or approaching 0.

• The (–) sign above the 5 represents that the fraction is approaching 0 from the negative side.

A limit can be used to evaluate the f(x) values as x approaches 1 (see right). This limit would express the shape of the graph as x approaches 1 from the right, or from the values greater than one (as denoted by the 1^+.)

As x approaches 1, or x→1, the value of the fraction becomes infinitely large, or unbounded positively, as 50 is being divided by an increasingly very small positive number; however, since the fraction is negative, it approaches negative infinity, or becomes unbounded negatively. The +5 is insignificant in this scale; thus, (see left).

For the that limit would express the shape of the graph as x approaches 1 from the left, or from the values less than one (as denoted by the 1^-), (see right).

The only difference is that the value of 50/(x-1) becomes infinitely negative, or unbounded negatively, however, as the fraction is negative, the value of the fraction becomes infinitely positive, or unbounded positively. Again, the +5 is insignificant; thus, (see left).