In this quiz, we are going to develop some intuition behind the “limiting process.” To do this:
Let us look at an example similar to the example we saw in the last quiz.
Let model the position of a mass on a spring. Suppose we want its “instantaneous velocity” at the time
To do this, let us look at the average velocity on the interval and examine what happens as we let approach 0, as in the last quiz.
Which of the following expressions involving correctly models the average velocity of the mass on the interval
The average velocity of a mass on a spring up to a time is
Similar to what we saw in the previous quiz, if we plug in 0 for , the expression evaluates to the indeterminate form, so we can't immediately find the value of the instantaneous velocity.
We can, however, look at the value that approaches as approaches
Using the code environment and graph below, investigate what happens to as gets closer and closer to zero.
Try, for example, , , and . What seems to be happening to
We saw that as gets closer and closer to the expression gets closer and closer to
We'll have a lot more to say about the details, but this is the basic idea behind saying “the limit as approaches of is 1.” Symbolically, we write
So far, we have worked with limits in relation to a function that models velocity. However, there is no reason we can't generalize to all kinds of functions. This generalization allows us to define the limit problem.
Limit Problem: How can we determine whether or not a general function approaches a specific value at And if it does, what is this value?
For most functions, limit computation is fairly straightforward, but there are a lot of interesting exceptions with weird behavior. Let's analyze one of them so we can get a better understanding of what a limit is.
Suppose we want to determine how the function behaves as approaches In other words, we wish to determine
Does the function approach a specific value near If so, what value is it?
Use the code environment and graph below to help out.
The function doesn't seem to approach any particular value as approaches When this happens, we say that does not exist.
Which of these arguments best explains this behavior?
A. As gets close to also gets close to 0. Therefore, when we take the reciprocal of , we get something that gets bigger and bigger, so it doesn't approach any particular number.
B. As gets close to oscillates between and and so does the reciprocal, so it never gets close to any particular number.
C. When you actually plug in you get which is undefined, so the limit does not exist.
D. As gets close to gets larger and larger. Since oscillates forever between and as its input gets large, the final result doesn't approach any particular value.
Unusual oscillatory behavior can lead to a non-existent limit. Is this pattern true in general?
Let's investigate another oddball example of similar type:
What happens to as approaches
Use the code environment and graph visualization below to help out.
A. It gets closer and closer to because even though is doing crazy things as gets small, we're also multiplying by and that's going to
B. It doesn't approach any particular number, because as we saw in the last question, the term oscillates between and as gets small.
C. It gets closer and closer to
To end this quiz, let's recap what we have done so far:
While the two examples seen were stranger than most, these functions give insight into the heart of what a limit actually is.
In the next quiz, we will look at some more examples to gain a better understanding of the limit process.