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Math  /  Calculus

Question3. lim⁡x→0+xxx\lim _{x \rightarrow 0^{+}} x^{x^{x}}

Studdy Solution

STEP 1

What is this asking? What happens to xx raised to the power of xx raised to the power of xx, as xx gets really close to zero from the positive side? Watch out! Zero to the power of zero is a tricky thing!
Don't fall into the trap of thinking it's automatically zero or one!

STEP 2

1. Investigate xxx^x
2. Determine xxxx^{x^x}

STEP 3

Let's first look at what happens to xxx^x as xx approaches 0 from the right.
We can write xxx^x as eln⁡(xx)e^{\ln(x^x)}, which simplifies to ex⋅ln⁡(x)e^{x \cdot \ln(x)}.
We're doing this because it's easier to analyze exponents when they're brought down.

STEP 4

Now, we need to figure out what happens to x⋅ln⁡(x)x \cdot \ln(x) as xx approaches 00.
This looks like a limit we can tackle!
We can rewrite this as ln⁡(x)1x\frac{\ln(x)}{\frac{1}{x}}.
As xx goes to 00 from the right, we get −∞∞\frac{-\infty}{\infty}, which is an indeterminate form.
This means we can use L'Hôpital's rule!

STEP 5

Taking the derivative of the top and bottom, we get 1x−1x2\frac{\frac{1}{x}}{-\frac{1}{x^2}}, which simplifies to −x-x.
As xx approaches 00, this goes to **0**.

STEP 6

So, x⋅ln⁡(x)x \cdot \ln(x) approaches **0** as xx approaches 00 from the right.
Therefore, xx=ex⋅ln⁡(x)x^x = e^{x \cdot \ln(x)} approaches e0=1e^{\textbf{0}} = \textbf{1} as xx approaches 00 from the right.
Awesome!

STEP 7

Now, let's put it all together.
We want to find lim⁡x→0+xxx\lim_{x \rightarrow 0^{+}} x^{x^{x}}.
We can rewrite this as lim⁡x→0+x(xx)\lim_{x \rightarrow 0^{+}} x^{(x^x)}.

STEP 8

From our previous investigation, we know that as xx approaches 00 from the right, xxx^x approaches **1**.

STEP 9

So, we have something that looks like xx raised to the power of something approaching **1**.
That means our limit becomes lim⁡x→0+x1\lim_{x \rightarrow 0^{+}} x^{\textbf{1}}, which is just lim⁡x→0+x\lim_{x \rightarrow 0^{+}} x.

STEP 10

And as xx approaches 00, the whole expression approaches **0**!

STEP 11

The limit of xxxx^{x^x} as xx approaches 00 from the right is **0**.

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