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 canβt see what is happening in the neighborhood of .
Figure1.A graph of a function that has not been rendered properly.
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 -value are you approaching?
Think of the same process, but this time from the right. Youβre falling down and to the left this time as you come close to the missing portion. What -value are you approaching?
When studying functions in algebra, we often focused on the value of a function given a specific -value. For instance, finding for some function . 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.
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.)
provided that we can make as close to as we like by taking sufficiently close (but not equal) to . If we cannot make as close to a single value as we would like as approaches , then we say that does not have a limit as approaches .
provided that we can make the value of as close to as we like by taking sufficiently close to while always having . We call the left-hand limit of as approaches . Similarly, we say is the right-hand limit of as approaches and write
The part of the theorem that starts with βSupposeβ¦β forms the assumptions of the theorem, while the part of the theorem that starts with βThenβ¦β is the conclusion of the theorem. What are the assumptions of the Squeeze Theorem? What is the conclusion?
The assumptions of the Squeeze Theorem can be restated informally as βthe function is squeezed between the functions and around .β Explain in your own words how the two assumptions result into a βsqueezing effect.β
Letβs 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.
Figure5.A pictorial example of the Squeeze Theorem.
Using trigonometry, one can show algebraically that for values close to zero. Moreover, (we say that cosine is a continuous function). Use these facts and the Squeeze Theorem, to find the limit .