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Use the given graph of $ f $ to find a number $ \delta $ such thatif $ \left| x - 1 \right| < \delta $ then $ \left| f(x) - 1 \right| < 0.2 $

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04:24

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Missouri State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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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The graph of$$f(x)=2-\…

in this problem. Were given a graph of F. And our graph of F looks something like this is X and Y. And we have a curve And here's one going up here over to here. And yes, up here, down to here, 0.7. This is 1.2 and under here Is 1.1 And this is 0.8 mm. And we're asked to find a number delta such that what's that? What If AB Survive X -1 is less than delta, then F of X -1. Absolute value Is less than 0.2. Okay, So delta is going to be the distance. We are from one either positive or negative distance. Right? So that X -1 is less than that distance. Okay, So from the graph from the data we have here, right, we can see that We're we've got a delta, either .3 or of one. Delta is .3 or .1 right, So the f x is one of those points, Then f of X must one is less than .2. But here's the problem. If I choose .3, then that means that I get to choose. And that means I can have X equals 1.3, Which means then that F of X -1, It's now greater than .2, isn't it? Because what will happen, I will end up with a point out here At 1.3. And if that curve continues on, I'm way down here. And so then this distance here is greater And 0.2. So what does that tell me? It tells me that only delta Equal to 0.1 works for this problem, so that's my number delta such that F of X -1 is always less than .2.

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