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Use a table of values to estimate the value of th…

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Problem 27 Easy Difficulty

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$ \displaystyle \lim_{x \to 0^+}x^x $


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02:47

Daniel Jaimes

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Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Related Topics

Limits

Derivatives

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Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

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Harvey Mudd College

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Lectures

Video Thumbnail

04:40

Limits - Intro

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.

Video Thumbnail

04:40

Derivatives - Intro

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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Watch More Solved Questions in Chapter 2

Problem 1
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Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55

Video Transcript

All right. We want to make a table of values to try to uh see if the limit of X to the X power as X approaches zero from the positive side uh exists and what it might be. So we're going to have a few X values that are approaching zero from the positive side. It's a 00.1 than a little closer to 0.1 than even closer 0.1. And then really close to 0.1. And so we want to calculate what is X to the ex uh the nexus 0.1, we're going to have 0.1 Raised to the .1 power. When next is .01, we're going to take .01 Raised that to the .01 power. Let me rewrite that. And some of these might get so small for the calculator that I might not even actually be able to calculate them. But for X equal 2.1 X dx 0.1 to the 0.1. So let me pause this video and see if I can get some of these numbers. Okay, so kind of surprising. Um I thought these would be really tiny numbers, but when I did 0.1 race to the 0.1 power, I got approximately 0.79 point oh one race at a point a one hour I got 10.95. So I thought they'd be such tiny numbers that calculator wouldn't even show it. Um but they're actually, you know, .95. The next 1.001 raised to the 0.1 hours 0.993. So we can do one more. Uh Let's do When X. is zero. The nexus .0001 X. To the X. is .0001 raised to the .0001 power. And if I put that in the calculator let's see what I get .0001 raised 2.0001. And I get .999. So it looks like as X is approaching zero from the positive side. It looks like the function values. Uh They look like they're approaching the value of one. So let's take a look at the graph of the function X. To the X. Power. And let's look what happens to that graph as we approach zero from the positive side. So here's a graph of X. T. X. And I put restriction on the X. Values X has to be greater than zero because X is approaching zero from the positive side. So it never actually become zero. It just gets close to zero from the positive side. And I kept X greater than zero when I graft to function X. D. X. Because it's not defined when X is zero. You would have 0 to 0 which is undefined. Um You can actually see that if you try to get this point right here, Okay affects 02 functions not even defined, but you can see that as X approaches zero from the positive side. Uh Let's look at the function values 00.7 point 74 Point 80.8 4 .9 .9 1 .929397. I see you can see um that as X approaches zero from the positive side, the function is approaching the value of one.

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Calculus: Early Transcendentals

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Related Topics

Limits

Derivatives

Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Joseph Lentino

Boston College

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

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.

Video Thumbnail

04:40

Derivatives - Intro

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.

Join Course
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