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Calculus Level 2

Find the following limit:

lim ⁡ x → 0 ( x x x − x x ) \Large \lim_{x\to 0} \left(x^{x^x}- x^x\right)


The answer is -1.

Hana Wehbi by Hana Wehbi , 51, USA

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3 solutions

L = lim ⁡ x → 0 ( x x x − x x ) Note that lim ⁡ x → 0 x x = 1 = 0 1 − 1 = − 1 \begin{aligned} L & = \lim_{x \to 0} \left(x^{x^x} - x^x \right) & \small \blue{\text{Note that }\lim_{x \to 0} x^x = 1} \\ & = 0^1 - 1 = \boxed {-1} \end{aligned}

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Hana Wehbi
Mar 2, 2020

lim ⁡ x → 0 x x = 1 \lim_{x\to 0} x^x = 1 Click HERE to see the proof.

lim ⁡ x → 0 x x x = 0 1 = 0 ⟹ \lim _{x\to 0} x^{x^{x}} = 0^1 = 0\implies

lim ⁡ x → 0 ( x x x − x x ) = lim ⁡ x → 0 x x x − lim ⁡ x → 0 x x = 0 − 1 = − 1 \lim_{x\to 0}( x^{x^{x}} - x^x ) = \lim_{x\to 0} x^{x^{x}} - \lim_ {x\to 0} x^x = 0 - 1 = -1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I got it right .

Nikola Alfredi - 1 year, 3 months ago

The limit should look like lim ⁡ x → 0 ( x x x − x x ) \lim_{x\to 0} (x^{x^x}-x^x) . Otherwise x x x^x will remain outside the limit

0 Helpful 0 Interesting 0 Brilliant 0 Confused

@Alak Bhattacharya Thank you, you were right, I fixed it.

Hana Wehbi - 1 year, 3 months ago

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