A Weird Limit

Calculus Level 3

lim ⁡ x → 1 ( x x sin ⁡ ( 1 − x ) ) = ? \Large \lim_{x \to 1} \left ( x^\frac{x}{\sin(1-x)} \right) = \, ?

1 e \frac{1}{e} ∞ \infty 1 1 0 0 − 1 -1

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Relevant wiki: Repeated Application of L'Hopital's Rule - Medium

L = lim ⁡ x → 1 x x sin ⁡ ( 1 − x ) = lim ⁡ x → 1 exp ⁡ ( ln ⁡ x x sin ⁡ ( 1 − x ) ) = lim ⁡ x → 1 exp ⁡ ( x ln ⁡ x sin ⁡ ( 1 − x ) ) A 0/0 cases, L’H o ˆ pital’s rule applies. = lim ⁡ x → 1 exp ⁡ ( ln ⁡ x + 1 − cos ⁡ ( 1 − x ) ) Differentiate up and down w.r.t. x . = e − 1 = 1 e \begin{aligned} L & = \lim_{x \to 1} x^{\frac x{\sin (1-x)}} \\ & = \lim_{x \to 1} \exp \left(\ln x^{\frac x{\sin (1-x)}}\right) \\ & = \lim_{x \to 1} \exp \left(\frac {x \ln x}{\sin (1-x)}\right) & \small {\color{#3D99F6}\text{A 0/0 cases, L'Hôpital's rule applies.}} \\ & = \lim_{x \to 1} \exp \left(\frac {\ln x + 1}{-\cos (1-x)}\right) & \small {\color{#3D99F6}\text{Differentiate up and down w.r.t. }x.} \\ & = e^{-1} = \boxed{\dfrac 1e} \end{aligned}

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Thank you for the nice solution.

Hana Wehbi - 4 years, 7 months ago

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