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

$ \displaystyle \lim_{t \to 2}\left( \frac{t^2 - 2}{t^3 - 3t + 5} \right)^2 $

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04:53

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Missouri State University

Campbell 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.

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to evaluate this limit. We first apply the limit of a power and so from here we have limit. As T approaches two of t squared minus two over T. To the third power minus three. T plus five. All of this race too. The second power. Next you want to apply the limit of a Kocian. And so we have limit as T approaches to of T squared minus two. This all over the limit S. T approaches to of T. To the third power minus three. T plus five. This race too, the second power. And then from here we apply limit of assam or difference. And so we have the square of the limit as T approaches two of T squared minus limit as T Approaches two of 2. This all over limit as T approaches to of T Q minus limit as T approaches to of three T plus limit as T approaches two of five. And then from here we apply limit of a constant and so we simplify this to the square of limit as T approaches to of T squared minus two over have limits S. D. Approaches to of T. To the third power minus limit as T approaches to of three T plus five. And then we apply limit of a power and so we have the square of limits As he approaches to of tea and then Square -2. All over. You have limit as T approaches to of T. And then Raised to the 3rd power minus three limit as the approaches to of T. And then plus five. And so we have two squared minus two over. Due to the 3rd power minus three times 2 plus five squared we have two over eight minus six plus five and then squared we have the square of 2/7 or that's four over 49. And so this is the value of the limits.

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