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Calculus Level 5

lim ⁡ x → ∞ [ ( ( f ( x ) + 1 ) ( 2 f ( x ) + 3 2 ) ( 4 f ( x ) + 1 ) ) 1 / 3 − f ( x ) f ( x ) ] x ( x − 2 f ( x ) ) \lim_{x\to\infty} \left [\dfrac{(( f(x) + 1) \left( 2f(x) + \dfrac32\right) (4f(x) + 1) )^{1/3} - f(x)}{f(x)} \right]^{x(x-2f(x))}

Let f : [ 0 , ∞ ) → [ − 4 , ∞ ) f: [0,\infty) \rightarrow [-4,\infty) be a function with an asymptote 2 y − x + 3 = 0 2y - x+3 = 0 . Compute the limit above.


The answer is 2980.95798.

Harshit Jain by Harshit Jain , 21, India

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1 solution

Otto Bretscher
Apr 14, 2016

The series expansion at ∞ \infty of the expression in the brackets starts with 1 + 4 3 y 1+\frac{4}{3y} so that the answer is lim ⁡ y → ∞ ( 1 + 4 3 y ) ( 2 y + 3 ) 3 = lim ⁡ u → ∞ ( 1 + 1 u ) 8 u = e 8 ≈ 2980.95798 \lim_{y\to\infty}\left(1+\frac{4}{3y}\right)^{(2y+3)3}=\lim_{u\to\infty}\left(1+\frac{1}{u}\right)^{8u}=e^8\approx \boxed{2980.95798} .

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