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Graph the function $ f(x) = \sin (\pi/x) $ of Example 4 in the viewing rectangle $ [-1, 1] $ by $ [-1, 1] $. Then zoom in toward the origin several times. Comment on the behavior of this function.

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No matter how many times we zoom in toward the origin, the graphs of $f(x)=\sin (\pi / x)$ appear to consist of almost-verticallines. This indicates more and more frequent oscillations as $x \rightarrow 0$

01:50

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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pain of pi over X. So here we have the horizontal X axis. And uh you might immediately notice that this function uh gets uh very rapidly oscillating up and down and up and down and up and down as you're moving in from X equals one Towards zero from the positive side. Uh and once again dysfunction oscillates. Um Here is function value of one on the y axis. Here's a function value of negative one on the y axis. So as X approaches zero from the negative side from the left, once again, this function is oscillating between one and negative one. Here to function, is he going to one again negative one, 1 -1. And it starts to do it a little bit more often and a little quicker and then to close X gets to zero from either side uh to function starts really oscillating really, really, really really really, really fast. We can do them in a little bit to kind of see that. So over here X was .2. Um so now excess .1, you can see that the functions still oscillating very quickly. And the closer you get closer x gets to 0 uh is going to keep oscillating but it's just going to do it more and more often. Uh so we zoom in a little bit here, you can see, you know, is continuing to oscillate quicker and quicker. Uh One thing very much worth noting is if you look at this function sign up i over X. It is not defined when X equals zero because you cannot divide uh by zero. So this function is not continuous when X equals zero, and it's very hard to tell. Um But you know, looking at this graph, but when X equals zero, uh there is no point on the graph, because to function is not defined there. Yeah.

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