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

$\lim _{x \rightarrow 3}\left(x^{2}+x-4\right)=8$

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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Prove the statement using …

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02:17

mhm Yeah So by definition we can just substitute the X with three in the situation. So you've got the meat Acts of Short 2, 3 Ex choir plus X -4 Just equals history Squire plus 3 -4 which equals to eight in the situation. That means for arbitrary It's a little larger than zero. Yeah we got delta larger than zero. Such that our function -8 is less than ε will never X -3 is less than Delta. That means you have X choir plus x minus chop off is less than epsilon. So if we take the factory a got tax minus three X-plus four in the absolute value last and excellent. So with the exception was close to three. We can assume that delta is less than one. Then we got x plus four And X -3. It's less than three class. Del top last four Holmes absolute value of X minus street. So this one is less than eight Times X -3. So in this situation we can just choose its plunge equals two. Uh their data equals two. Excellent. Over eight

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