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Evaluate the limit and justify each step by indicating the appropriate properties of limits.

$ \displaystyle \lim_{x \to \infty} \dfrac{2x^2 - 7}{5x^2 + x -3} $

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$\frac{2}{5}$

03:30

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

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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So here we are given a specific limit and were asked to evaluate the limits as X approaches infinity of two, X squared minus seven over five X squared plus x minus three. So as X approaches large numbers, relatively speaking compared to our constant terms, the X axis, The X values in this would be significantly greater. So you can ignore our constant terms. Also, X squared would also be significantly greater than just X for large numbers so we can ignore X. And as a result we can find that our limit would be the limit as X approaches infinity of two, X squared divided by five X. Work. So we can cancel out this X squared And we can find that we just have the limit as X approaches infinity of a constant term. So this would just be equivalent to 2/5. We can also evaluate this by our other method for evaluating limits. Lockable rule. So this would just be taking the derivative of the numerator and the derivative of the nominator. But we require multiple iterations in this case. So the first iteration before X. This would be 10, X plus one. Then we will need another iteration, this would become four and this would become 10 And once again this would be the limit as X approaches infinity for over 10 which is just equal equivalent to 2/5 and simplest form. And this gifts our final answer

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