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Prove, using Definition 6, that $ \displaystyle \lim_{x \to -3} \frac{1}{(x + 3)^4} = \infty $.

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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 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.

03:16

Prove the limit statements…

this's problem or forty two of the Stuart Calculus, either edition section two point four prove using definition six. That the limit is experts is negative. Three of the function one over the quantity X plus three to the fourth power. It's called an identity. So definition six states that for any m greater than zero there exists a delta grid. Isn't zero such that if the absolute value of the difference between X and me is Liston daughter, then the function is guaranteed to be great with them. So we just being with second inequality and see what relationship we confined between them and Delta. So our function this one over the quantity X plus three to the fourth and we want to guarantee that this greater than any value we choose for them. We re arrange this one over him created than the quantity express T to the fourth. And then we take ah, the forth route to put science And here the forth route cancel. And what we're left with is that the quantity extra story must be less Stan the must be listen one over the forth route of him. If we recall the conditional for Delta Is that expectancy or in this case, Xmas Negative three. Must be. This difference was fearless and Delta and And we re write it this way. You can see that Delta and this term here. Ah, having value equal to that is an appropriate choice that guarantees bad for any delta. Are you can find any delta for any given value? Mm. For example. Mmm is ten thousand. The four three ten thousand is ten. So choosing the delta equal to one over ten. Ah provide prevents the conditions needed to prove this limit. So for any value, them there exists a daughter, and therefore the positive values and the both correspond to these inequalities. And everything is consistent there forthis summit as expressing it. Three of dysfunction, unequal to infinity.

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