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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 $

Answer

$$\begin{aligned}
&\text { For } f(x)=x^{x}\\
&\begin{array}{|l|c|}
\hline x & f(x) \\
\hline 0.1 & 0.794328 \\
0.01 & 0.954993 \\
0.001 & 0.993116 \\
0.0001 & 0.999079 \\
\hline
\end{array}
\end{aligned}$$
It appears that $\lim _{x \rightarrow 0^{+}} f(x)=1$ The graph confirms that result.

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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.