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Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit.

$ \displaystyle \lim_{x \to 10}(3 - \frac{4}{5}x) = -5 $

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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This is problem number twenty of the Stuart Calculus eighth edition, Section two point four Prove the statement using the absolute out the definition of a limit. No limit as X approaches ten of the function three minutes for fifty X is equal to negative five and the Epsilon Delta definition of a limit Is that reverie? Absolute Greatest zero. We must find a delta that is created in the room. Ah, such that the following two inequalities are true. One dependent on the other. If the outside value of the difference between X and A is listing delta, then the value of the difference between the function on the limit is less than epsilon. So we are going to use the second inequality start with here and use our function that was given three minus forfeit. Tex, subject our limit. L is equal to taking a five western Absalom and we simplify here combined the like terms three minus thing. If I was eight. Thank you, Mrs Negative for Fitz X plus eight Weston Salaam. One step that we can take is to change this eight into a fraction with denominator of five before five x plus. Already over five that's equal to eight are less than absolute. If we combine, the numerator is here and they get for expose forty. No, that's the numerator, the denominators five. And at this point, we can factor out a negative for fifths. So we're gonna have is the absolute value. Negative forfeits factored out and it leaves us with X plus ten. Sorry, X minus ten. They also have that. Listen, Absalon, this absolutely cancels the negative s. So it's just forfeits. And then we want other five fourths to both sides and that gets us find Absalon or four. So we've gone down to that point. We recall this. First Inequality X minus A is ten in our case in this limit ten less than dull term. And we see the similarities between these two final inequalities and we see that we can find a delta For every Absalon provided that the maximum value that doctor is five absolute or for you can choose anybody. Listen, that. But this is sufficient enough to prove our statement

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