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Handout Daily Prep 1.2 - The Notion of a Limit

Section Overview

We will now begin to investigate the idea of the limit of a function at a given input value. Limits are important in calculus for many reasons, but perhaps most important for how they allow us to formally connect instantaneous velocity to average velocity.
This section covers the following concepts: Definition and notation of limits. Graphical interpretation, table of values, algebraic computation. Indeterminate form of type \(0/0\text{.}\)

Section Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
  • State the two related formulas for the average velocity of an object with position function \(s\) on an interval \([a,b]\) and on the interval \([a,a+h]\text{.}\)
  • Interpret the meaning of notation such as “\(\displaystyle \lim_{x\rightarrow 2}f(x) =-3\text{.}\)” Use limit notation.
  • Use computations of average velocity to determine instantaneous velocity.
  • Provide examples (both graphically and numerically) of functions that do not have a limit at a specific value.
  • Estimate/hypothesize the limit of a function at a point on a graph (or determine that it does not exist).

Section To prepare for class

Complete all actions listed below. Respond to the questions highlighted with Submit.
A single .pdf should be uploaded to D2L Brightspace. Strive to use valid notation and correct mathematical language and syntax. All answers should be briefly justified, whether justification is specifically requested or not.
  • Read the beginning of section 1.2 (up until the preview activity).
  • 📝 [Submit] Do Preview Activity 1.2.1.
  • Read section 1.2.2: The Notion of Limit.
  • Watch this video on evaluating limits from the graph of a function: Limits (6:01).
  • 📝 [Submit] Do Example 1.2.4(a) in the textbook (without reading the solution), using the graph of the function: Use Desmos to graph \(f\) (you define \(f\) by typing \(f(x) = (4-x^{2})/(x+2)\text{,}\) and you can click and drag points on the graph to see the precise coordinates). Finally, read the solution. Is the graph that Desmos produces accurate? Explain. Produce a screen capture of your work in doing Example 1.2.4(a) using Desmos.
  • Watch this video on evaluating limits using graphing tools (GeoGebra): Limits of Functions Using Graphing Tools (6:26).
  • 📝 [Submit] Redo Example 1.2.4(a) from the textbook using GeoGebra rather than Desmos. Produce a screen capture of your work in doing Example 1.2.4(a) using GeoGebra.
  • Do the following exercises.
    1. Are the following statements about \(f\) shown in Figure 15 true or false? Why?
      Graph of a function on a coordinate grid. The x-axis is horizontal and the y-axis is vertical. On the left, a smooth curve increases to an open circle near x = −4, then decreases to a filled point at x = 0. From x = 0 to about x = 2, the graph rises from an open circle, then transitions into a rapidly oscillating curve between x ≈ 3 and x ≈ 5. To the right of this, the graph becomes smooth again, rising to a peak near x = 7 and then decreasing.
      Figure 15.
      1. \(x=a\) is in the domain of \(f\)
      2. \(\displaystyle \lim_{x \rightarrow a}f(x)\) exists
      3. \(\displaystyle \lim_{x \rightarrow a^+}f(x)\) is equal to \(\displaystyle \lim_{x \rightarrow a^-}f(x)\)
    2. Sketch the graph of an example of a function \(f\) that satisfies all of the following: \(\displaystyle \lim_{x \rightarrow 0}f(x) = 1\text{,}\) \(\displaystyle \lim_{x \rightarrow 3^-}f(x) = -2\text{,}\) \(\displaystyle \lim_{x \rightarrow 3^+}f(x) = 2\text{,}\) \(f(0) = -1\text{,}\) and \(f(3) = 1\text{.}\)
    3. For the function \(f\) whose graph is given in Figure 16, state the value of each quantity, if it exists. If it does not exist, explain why.
      Graph of a function on a coordinate grid with the horizontal axis labeled x and the vertical axis labeled y, both marked with integer tick marks. On the far left, the graph begins below the x-axis and increases to an open circle near x equals minus four at a y-value slightly above two. From that open circle, the graph curves downward to a filled point at x equals zero with y equal to zero. Near x equals zero, a second branch begins at an open circle below the x-axis. From this point, the graph increases smoothly, passing through an open circle near x equals two at a y-value slightly above one. Just to the right of x equals two, the graph enters a region of rapid oscillation, with the y-values fluctuating repeatedly between positive and negative values over a short interval. For x greater than about four, the oscillations stop and the graph becomes smooth again, rising to a local maximum near x equals six and then decreasing slightly toward the right edge of the graph. The figure shows open circles indicating points where the function is not defined and illustrates smooth behavior, discontinuities, and oscillatory behavior on different intervals.
      Figure 16.
      1. \(\displaystyle \displaystyle \lim_{x \rightarrow -3^-}f(x)\)
      2. \(\displaystyle \displaystyle \lim_{x \rightarrow -3^+}f(x)\)
      3. \(\displaystyle \displaystyle \lim_{x \rightarrow -3}f(x)\)
      4. \(\displaystyle f(-3)\)
      5. \(\displaystyle \displaystyle \lim_{x \rightarrow 0^-}f(x)\)
      6. \(\displaystyle \displaystyle \lim_{x \rightarrow 0^+}f(x)\)
      7. \(\displaystyle \displaystyle \lim_{x \rightarrow 0}f(x)\)
      8. \(\displaystyle f(0)\)
      9. \(\displaystyle \displaystyle \lim_{x \rightarrow 2}f(x)\)
      10. \(\displaystyle f(2)\)
      11. \(\displaystyle \displaystyle \lim_{x \rightarrow 5^+}f(x)\)
      12. \(\displaystyle \displaystyle \lim_{x \rightarrow 5^-}f(x)\)

Checkpoint 17. Limits versus Function Values.

Section After class

Solidifying the concepts discussed in class through practice is necessary to build your skills. Mathematics is not a spectator sport!

Section Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
  • Estimate the limit of a function at a point using the algebraic formula (or determine that it does not exist). This includes indeterminate forms of type \(0/0\text{.}\)
  • Compute instantaneous velocities using both formulas.
  • Demonstrate instantaneous velocity graphically on a position graph.
  • Explain how instantaneous velocity is thought of as a limit of average velocities.

Section Additional suggestions

Section Answers

    1. True; the function is defined (as per black dot)
    2. True; the limits from the left and the right are the same
    3. True; however, these limits do not equal \(f(a)\)
  2. Answer will vary.
    Detailed graph of a piecewise-defined function on a square coordinate grid. The horizontal axis is labeled x and the vertical axis is labeled y, with integer tick marks shown. On the left side of the graph, for x less than 0, a smooth decreasing curve approaches the point (0, 1), ending at an open circle at (0, 1). At x = 0, the function instead has a filled point at (0, −1). From x = 0 to x = 3, the graph is a straight line with negative slope, starting at the open circle at (0, 1) and descending to an open circle at (3, −2). At x = 3, there is a filled point at (3, 1), distinct from the open points on the other pieces. For x greater than 3, the graph is a straight line with positive slope, beginning at an open circle at (3, 2) and increasing to the right. The graph emphasizes discontinuities at x = 0 and x = 3, shown by open circles and filled points at the same x-values but different y-values.
    Figure 18. One possible graph.
    1. \(\displaystyle \displaystyle \lim_{x \rightarrow -3^-}f(x) = 4\)
    2. \(\displaystyle \displaystyle \lim_{x \rightarrow -3^+}f(x) = 4\)
    3. \(\displaystyle \displaystyle \lim_{x \rightarrow -3}f(x) = 4\)
    4. \(f(-3)\) does not exist
    5. \(\displaystyle \displaystyle \lim_{x \rightarrow 0^-}f(x) = 1\)
    6. \(\displaystyle \displaystyle \lim_{x \rightarrow 0^+}f(x) = -1\)
    7. \(\displaystyle \lim_{x \rightarrow 0}f(x)\) does not exist
    8. \(\displaystyle f(0)= 1\)
    9. \(\displaystyle \displaystyle \lim_{x \rightarrow 2}f(x) = 2\)
    10. \(f(2)\) does not exist
    11. \(\displaystyle \displaystyle \lim_{x \rightarrow 5^+}f(x) = 3\)
    12. \(\displaystyle \lim_{x \rightarrow 5^-}f(x)\) does not exist
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