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5.2.1 Introduction to Limits

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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.

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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? _____

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

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5-43 Look at your graph of f(x) = + 3 As x gets larger and larger, f(x) gets closer and closer to 3.

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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.

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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.

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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.”

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