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Math 1241, Spring 2014 Section 3.1, Part One

Math 1241, Spring 2014 Section 3.1, Part One. Introduction to Limits Finding Limits From a Graph One-sided and Infinite Limits. Conceptual idea of a Limit. If I live close enough to campus, I can drive there in a very short amount of time.

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Math 1241, Spring 2014 Section 3.1, Part One

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  1. Math 1241, Spring 2014Section 3.1, Part One Introduction to Limits Finding Limits From a Graph One-sided and Infinite Limits

  2. Conceptual idea of a Limit • If I live close enough to campus, I can drive there in a very short amount of time. • Intuitively, this is a true statement. However, it’s somewhat ambiguous. Why? • What do we mean by… • “close enough” to campus? • a “very short” amount of time?

  3. Conceptual idea of a Limit • “Close enough to campus” should mean “within a certain distance.” What distance? • “A short amount of time” should mean “less than a certain amount of time.” How much? • We must also put these together so that the following statement is true: If I live within ____ miles of campus, I can drive there in less than ____ minutes.

  4. Conceptual idea of a Limit • In theory, my commute time is determined by (is a function of) the distance I live from campus. Reality is more complicated, but… • If I specify my maximum commute time, could you determine my maximum distance? • Could you do this regardless of what maximum time I specify? • These questions are related to the precise, mathematical definition of a limit.

  5. Distance versus Travel Time

  6. A more complicated limit • Google Maps: It takes 18 minutes to drive 15.5 miles from Turner Field to Clayton State (obviously, this ignores downtown traffic). • So…. If I start “close enough” to Turner Field, my driving time to Clayton state is “nearly” 18 minutes? • To make this precise, what would you need to specify? (Answers on the next slide)

  7. A more complicated limit • You would need to tell me: • Within what distance of Turner Field? • How close to 18 minutes? • If I start within ____ miles of Turner Field, my drive time to CSU is within ____ minutes of the 18 minutes claimed by Google Maps. • Question: If my drive time is nearly 18 minutes, did I start close to Turner Field?

  8. Distance versus Travel Time

  9. An easy algebraic limit • In general, we’ll have a function y = f(x), and ask what happens to the output (y) as the input (x) gets “close to” some fixed value (a). • Example: What happens to the value of the function y = 2x - 3 as x gets close to 2? • Try to answer this without plugging in x = 2. The reason for this restriction will become clear in later examples. • This is a straight line, try drawing a graph!

  10. Example: f(x) = 2x - 3

  11. Another Example • What happens to the value of the function as x gets close to the value of 2? • In this case, we cannot simply plug in x = 2. • However, if x is not equal to 2, the above expression can be simplified algebraically. • Alternatively, we can draw a graph.

  12. Limit Notation • We use the following set of symbols: • Read this as, “The limit, as x approaches a, of f(x) is equal to L.” • Informally, this means: If the value of x is “close enough” BUT NOT EQUAL to a, then the value of y is “close” (possibly equal) to the number L. • In the previous examples, we had:

  13. Graphical Exercise For the function f(x) shown to the right, find… • = _____ • = _____ • = _____ • = _____ Pay attention to the open and closed dots!

  14. Some notes about limits • The limit of a function must be a single number. This means a particular limit might not exist • Previous example: No limit as x approaches -4. • You can often (BUT NOT ALWAYS) evaluate the limit of a “simple” function by plugging in the value x = a. We will discuss when this is permissible in Section 3.2 (Continuity). • Although we’ll avoid the formal definition of a limit, but we will introduce algebraic rules for evaluating limits (next time).

  15. One-sided limits • Question: What is the value of the following? • Note that we cannot plug in x = 0. It may be helpful to draw a graph. • Fill in the blank: The value of f(x) is close to _____ whenever x is close to 0.

  16. = ??

  17. One-sided limits • We can use the following notation: • The first is a left-sided limit. As x approaches 0 from the left, the function value is close to -1. • The second is a right-sided limit. As x approaches 0 from the right, the function value is close to 1. • When the left and right limits are not equal, the ordinary, two-sided limit DOES NOT EXIST.

  18. Infinite Limits • If x is close to zero, then the function is close to what number? Here is the graph:

  19. Infinite Limits • IMPORTANT: DOES NOT EXIST!!! • There is a reason for this. As x approaches 0, the function value keeps getting larger, and never approaches any particular value. • Notation: • But you CANNOT treat the infinity symbol as though it were an ordinary number.

  20. Examples of Infinite Limits Convince yourself (possibly by drawing a graph) that the following are true: For the left-sided limit, the means that the function value continues to decrease, and does not approach any particular value.

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