Touching an Asymptote

Calculus Level 3

lim ⁡ x → a 21 x − sin ⁡ ( 7 π x ) = lim ⁡ x → ∞ 21 x − sin ⁡ ( 7 π x ) \lim_{x\to a} \dfrac{21}{x} - \sin\left(\dfrac{7\pi}{x}\right) = \lim_{x\to\infty} \dfrac{21}{x}-\sin\left(\dfrac{7\pi}{x}\right)

Find the positive value of a a , satisfying the equation above.

Inspired by Round & Round .


The answer is 42.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

The RHS lim ⁡ x → ∞ 21 x − sin ⁡ 7 π x = 0 − 0 = 0 \displaystyle \lim_{x \to \infty} \frac {21}x - \sin \frac {7\pi}x = 0 - 0 = 0 . Therefore the LHS = 0 =0 and ⟹ sin ⁡ 7 π x = 21 x \implies \sin \dfrac {7\pi}x = \dfrac {21}x . The obvious solution is x = 42 x=\boxed{42} , ⟹ sin ⁡ 7 π 42 = 21 42 \implies \sin \dfrac {7\pi}{42} = \dfrac {21}{42} ⟹ sin ⁡ π 6 = 1 2 \implies \sin \dfrac \pi 6 = \dfrac 12 , which is true.

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