Tetration to infinity

Calculus Level 3

ϕ = lim ⁡ n → ∞ ( lim ⁡ x → 0 ( ( sin ⁡ x ) ( sin ⁡ x ) ( sin ⁡ x ) ⋯ ⏞ n times x x x ⋯ ⏟ n times ) ) \phi = \lim_{n\to\infty} \bigg(\lim_{x\to 0} \bigg(\dfrac{\overbrace {\,(\sin x)^{\,(\sin x) ^{\,(\sin x)^{\cdots}}}}^{\text{n times}}}{\underbrace{x^{x^{x^{\cdots}}}}_{\text{n times}}}\bigg)\bigg)

Find the limit above.

This is an original problem.

1 0 Limit doesn't exist ∞ \infty

Naren Bhandari by Naren Bhandari , 21, India

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1 solution

Parth Sankhe
Oct 10, 2018

sin ⁡ x = x \sin x = x when x tends to 0. As long as no addition or subtraction is involved, the sin in sinx can be removed to give just x. Thus the numerator and the denominator cancel out giving the answer as 1

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