What is the important term?

Calculus Level 2

lim ⁡ x → ∞ ( e 11 x − 7 x ) 1 3 x \large \lim_{x\to \infty} (e^{11x} - 7x)^{\frac1{3x}}

If the limit above equals to e a / b e^{a/b} for coprime positive integers a , b a,b , find the value of a + b a+b .


The answer is 14.

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Rohan Bansal by rohan bansal , 22, India

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1 solution

lim ⁡ x → ∞ ( e 11 x − 7 x ) 1 3 x = lim ⁡ x → ∞ e ln ⁡ ( e 11 x − 7 x ) 3 x \lim_{x\to \infty} (e^{11x}-7x)^{\frac{1}{3x}}=\lim_{x\to \infty} e^{\frac{\ln(e^{11x}-7x)}{3x}} = e lim ⁡ x → ∞ 11 e 11 x − 7 3 ( e 11 x − 7 x ) = e 11 3 =e^{\displaystyle \lim_{x\to \infty} \frac{11e^{11x}-7}{3(e^{11x}-7x)}}=e^{\frac{11}{3}}

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