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How close to -3 do we have to take $ x $ so that$$ \frac{1}{(x + 3)^4} > 10,000 $$

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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

University of Michigan - Ann Arbor

Idaho State University

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.

01:13

$$|x-3|^{3 x^{2}-10 x+…

01:09

Use $$(1+x)^{1 / 3…

02:07

$$3 x^{4}-10 x^{2}+3&g…

00:53

If $2^{-3 x}=\frac{1}{1000…

01:17

02:39

For $x \neq 0,$ how close …

02:27

$$(3 x-2)(x-3)^{3}(x+1…

01:54

Solve the exponential equa…

this problem forty one of this tour calculus In addition, section two point four How close to negative three Do we have to take X so that one divided by the quantity X plus three to the fourth power is greater than ten thousand? First we'Ll solve this algebraic lee we'LL divide by ten thousand on about science while also multiplying by the quantity Extra story to thie fourth power They won't take the forth route on both sides Here we'LL have X plus three that quantity to the fourth to the one fourth power Right, So we're doing one fourth power to vote science so that we can cancel out forthe here on the left We should be really able to reduce system into one tent because ten to the fourth power is ten thousand on the red side. We want to assure that this quantity as illustrious positive So we write it as the absolute value of X mystery. This specifically is the distance between value's X and negative three. And when we see that it must be values Liston one ten toe or point one And so that's how close we have to be and we can also do this graphically. Take this function one over the quantity X plus three to the fourth power and plotted against ten thousand to see where this across the function. And we see in this case that our heartbreak solution is correct within a point one anything less guarantees a value greater than ten thousand on exactly as zero point one is where I won't get the exact value of ten thousand. So the answer is that we have to be closer than zero point one in order for the statement to be true.

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