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Section 1.1 Limits Graphically (LT1)

Subsection 1.1.1 Activities

Activity 1.1.1.

In Figure 1 the graph of a function is given, but something is wrong. The graphic card failed and one portion did not render properly. We can’t see what is happening in the neighborhood of x=2.
Figure 1. A graph of a function that has not been rendered properly.
(a)
Imagine moving along the graph toward the missing portion from the left, so that you are climbing up and to the right toward the obscured area of the graph. What y-value are you approaching?
  1. 0.5
  2. 1
  3. 1.5
  4. 2
  5. 2.5
(b)
Think of the same process, but this time from the right. You’re falling down and to the left this time as you come close to the missing portion. What y-value are you approaching?
  1. 0.5
  2. 1
  3. 1.5
  4. 2
  5. 2.5

Activity 1.1.2.

In Figure 2 the graphic card is working again and we can see more clearly what is happening in the neighborhood of x=2.
Figure 2. A graph of a function that has rendered properly
(b)
What is the y-value that is approached as we move toward x=2 from the left?
  1. 0.5
  2. 1
  3. 1.5
  4. 2
  5. 2.5
(c)
What is the y-value that is approached as we move toward x=2 from the right?
  1. 0.5
  2. 1
  3. 1.5
  4. 2
  5. 2.5

Remark 1.1.3.

When studying functions in algebra, we often focused on the value of a function given a specific x-value. For instance, finding f(2) for some function f(x). In calculus, and here in Activity 1.1.1 and Activity 1.1.2, we have instead been exploring what is happening as we approach a certain value on a graph. This concept in mathematics is known as finding a limit.

Activity 1.1.4.

Based on Activity 1.1.1 and Activity 1.1.2, write your first draft of the definition of a limit. What is important to include? (You can use concepts of limits from your daily life to motivate or define what a limit is.)

Definition 1.1.5.

Given a function f, a fixed input x=a, and a real number L, we say that f has limit L as x approaches a, and write
limx→af(x)=L
provided that we can make f(x) as close to L as we like by taking x sufficiently close (but not equal) to a. If we cannot make f(x) as close to a single value as we would like as x approaches a, then we say that f does not have a limit as x approaches a.

Activity 1.1.6.

Figure 3. A piecewise-defined function
What is the limit as x approaches 0 in Figure 3?
  1. The limit is 1
  2. The limit is -1
  3. The limit is 0
  4. The limit is not defined

Definition 1.1.7.

We say that f has limit L1 as x approaches a from the left and write
limxβ†’aβˆ’f(x)=L1
provided that we can make the value of f(x) as close to L1 as we like by taking x sufficiently close to a while always having x<a. We call L1 the left-hand limit of f as x approaches a. Similarly, we say L2 is the right-hand limit of f as x approaches a and write
limx→a+f(x)=L2
provided that we can make the value of f(x) as close to L2 as we like by taking x sufficiently close to a while always having x>a.

Activity 1.1.8.

Refer again to Figure 3 from Activity 1.1.6.
(a)
Which of the following best matches the definition of right and left limits? (Note that DNE is short for "does not exist.")
  1. The left limit is -1. The right limit is 1.
  2. The left limit is 1. The right limit is -1.
  3. The left limit DNE. The right limit is 1.
  4. The left limit is -1. The right limit DNE.
  5. The left limit DNE. The right limit DNE.
(b)
What do you think the overall limit equals?
  1. The limit is 1
  2. The limit is -1
  3. The limit is 0
  4. The limit is not defined

Activity 1.1.9.

Consider the following graph:
Figure 4. Another piecewise-defined function
(a)
Find limxβ†’βˆ’3βˆ’f(x) and limxβ†’βˆ’3+f(x).
(b)
Find limxβ†’βˆ’1βˆ’f(x) and limxβ†’βˆ’1+f(x).
(e)
For which x-values does the overall limit exist? Select all. If the limit exists, find it. If it does not, explain why.
  1. βˆ’3
  2. βˆ’1
  3. 2
  4. 4

Activity 1.1.10.

Sketch the graph of a function f(x) that meets all of the following criteria. Be sure to scale your axes and label any important features of your graph.
  1. limxβ†’5βˆ’f(x) is finite, but limxβ†’5+f(x) is infinite.
  2. limxβ†’βˆ’3f(x)=βˆ’4, but f(βˆ’3)=0.
  3. limxβ†’βˆ’1βˆ’f(x)=βˆ’1 but limxβ†’βˆ’1+f(x)β‰ βˆ’1.

Activity 1.1.12.

In this activity we will explore a mathematical theorem, the Squeeze Theorem (Theorem 1.1.11).
(a)
The part of the theorem that starts with β€œSuppose…” forms the assumptions of the theorem, while the part of the theorem that starts with β€œThen…” is the conclusion of the theorem. What are the assumptions of the Squeeze Theorem? What is the conclusion?
(b)
The assumptions of the Squeeze Theorem can be restated informally as β€œthe function g is squeezed between the functions f and h around a.” Explain in your own words how the two assumptions result into a β€œsqueezing effect.”
(c)
Let’s see an example of the application of this theorem. First examine the following picture. Explain why, from the picture, it seems that both assumptions of the theorem hold.
Figure 5. A pictorial example of the Squeeze Theorem.
(d)
Match the functions f(x),g(x),h(x) in the picture to the functions cos⁑(x),1,sin⁑(x)x.
(e)
Using trigonometry, one can show algebraically that cos⁑(x)≀sin⁑(x)x≀1 for x values close to zero. Moreover, limxβ†’0cos⁑(x)=cos⁑(0)=1 (we say that cosine is a continuous function). Use these facts and the Squeeze Theorem, to find the limit limxβ†’0sin⁑(x)x.

Subsection 1.1.2 Videos

Figure 6. Video for LT1

Subsection 1.1.3 Exercises