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(a) Use a graph of $$ f(x) = \left( 1 - \frac{2}{x} \right)^x $$ to estimate the value of $ \displaystyle \lim_{x \to \infty} f(x) $ correct to two decimal places.(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

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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 is prime number twelve, the sewer calculus, eighth division, section two point six party use. A graph of function F is equal to quantity. One minus two of rex, razed to the X power to estimate the value of the limit has exported divinity of this function after correct two decimal places. So if we are, we're deployment this function one minus two wrecks that quantity race to the X power. We will get this function here where it would level off, had a certain value. And if we were to trace along dysfunction and see what value that was equal to, we see that its approach is approximately, sir point one three five on DSO. We would say that this limit is approximately zero point one four two two decimal places and that's our answer from using a graph. Isn't stable values for this function as to meet the limited for two small places? So with table vise, we should be able to choose a numbers large enough that we can see the trend of the function and how much it decreases by and where it seems to be approaching for very, very large numbers. We see that this function approaches approximately point one three five, three, two, two, four two Ford Decimal places on. And so we would say that our limit is more accurately equal to zero point one three five, and that is our final answer.

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