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Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{x \to 8}(1 + \sqrt[3]{x})(2 - 6x^2 + x^3) $

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04:30

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Baylor University

University of Nottingham

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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to develop this limit. We begin by applying the limit of a product. And so we have The limit as x approaches eight of one plus the cube rid of X. This times The limit as x approaches eight of 2 -6 x squared plus x rays to the third power Next you want to apply the limit of a sum and difference. And so we have The limit as x approaches eight of 1 plus The limit as x approaches eight of the key road of X. These times the limit as X approaches eight of two minus the limit as X approaches eight of 6 x squared Plus, we have the limit as x approaches eight of x rays to the third power. And then from here we want to apply the LTD a constant our route and power. And so from here we have one plus the cube root of the limit of X As X approaches eight and in these times We have two You have six times the limit of X as X approaches eight and then squared plus we have the limit as X approaches aid of x rays to the third power And so we have one plus to rid of eight. This times 2 -6 times it's squared plus we have eight raised to the 3rd power and so simplifying. We have one plus two times 2 -384 Plus 512. This will give us a value of 390 and so this is the value of the limits

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