What is any number raised to the power of infinity?

Number Theory Level 1

If 2 ∞ = ∞ 2^{\infty}=\infty , how much is ( 1 2 ) ∞ (\frac{1}{2})^{\infty} ?


The answer is 0.

P0156 M by p0156 M , 19, USA

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1 solution

Ved Pradhan
Jun 19, 2020

lim ⁡ x → ∞ 2 x = ∞ \lim_{x\rightarrow \infty} 2^{x} = \infty

lim ⁡ x → ∞ 1 2 x = \lim_{x\rightarrow \infty} \dfrac{1}{2}^{x} = lim ⁡ x → ∞ 1 2 x = \lim_{x\rightarrow \infty} \dfrac{1}{2^{x}} = lim ⁡ x → ∞ 1 lim ⁡ x → ∞ 2 x = \dfrac{\lim_{x\rightarrow \infty} 1}{\lim_{x\rightarrow \infty} 2^{x}} = = 0 = \boxed{0}

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