5.2.1Introduction to Limits

Limits• Today you will follow the travels of Benny and Bertha Bug.

With their help, we will look at graphs of rational functions and piecewise functions from a bug’s eye view to help convey the important concept of limits in an intuitive way.

5-41. Sketch the graph of f(x) = + 3.

Place a bold dot on the point of the graph corresponding to x = 5.

Benny Bug starts at this location and crawls along the curve, moving to the right.

If he keeps going in this direction, what y-value does Benny think he’s getting closer to? _____

As Benny keeps going farther and farther to the right, how low does Benny think he will get? _____

5-42. Consider the graph below when answering the questions below.

5-43

Look at your graph of f(x) = + 3

As x gets larger and larger, f(x) gets closer and closer to 3.

MATH NOTES - Definition of a One-Sided Limit

• The formal definition of a limit is well beyond this course. If you take engineering or calculus, you will get it there. For now you need to understand the basic notation.

• We say x → c+ if the values of x get closer and closer to c from the right; that is, x > c. Similarly, x → c− if the values of x get closer and closer to c from the left.

MATH NOTES - Definition of a One-Sided Limit

• If f(x) gets closer and closer to a given number L as x → c+, we say, “The limit of f(x) as x goes to c from the right is L.” This is written as .

• If f(x) grows without bound we say .

• If f(x) gets closer and closer to a number M as x → c−, we write .

MATH NOTES - Definition of a One-Sided Limit

You should know that although most college textbooks give ∞ as a limit (as shorthand for f(x) growing without bound), some books say “no limit.”