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Guess the value of the limit $$ \lim_{x \to \infty} \frac{x^2}{2^x} $$ by evaluating the function $ f(x) = x^2/2^x $ for $ x $ = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of $ f $ to support your guess.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Campbell University

Harvey Mudd College

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.

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this problem. Number eleven of this to recant his eighth edition section two point six. Guess the values the limit. The limit as X approaches infinity as that of the function X word or two to six by evaluating the function have for X equals zero one, two, three, four, five, six, seven, eight, nine, ten, twenty, fifty and a hundred. Then use the graph of F to support your guests so well, go ahead and calculate for each of these X's values the value of F on DH here in a spreadsheet. We have done just that, using dysfunction X squared over two x two decks. We see that for zero is zero, and then there's a slight increase in the function towards one. And then one point won't to five at a value Mexico's three and then back down to one frank sickles for, And then we see a steady decrease until it reaches a very small, small value very close to zero. So at this point, the limit as X approaches, infinity is what dysfunction is approaching. As we continually increase air about you, we have increased it alway two hundred and we see the value is already decreased a very, very small number close to zero. So we will guess that the function approaches zero as X approaches infinity. Therefore, this limit is equal to zero. Ah, and if we were to plot this craft to confirm our guests, we see that this function Although it has an increase initially overall, it will have that horizontal ask himto for y equals zero meaning that it will level off and approach zero not necessarily equally room, but for the limit. We do say that the limit does equal zero sex approaches infinity.

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