It ain't just about polynomials

Calculus Level 3

lim ⁡ x → ∞ 2 + 2 x + sin ⁡ ( 2 x ) ( 2 x + sin ⁡ ( 2 x ) ) ( e sin ⁡ ( x ) ) \large \displaystyle \lim_{x \to \infty} \cfrac{2+2x + \sin(2x)}{(2x + \sin(2x))(e^{\sin(x)})}

What is the value of the limit above?

None of these choices 1 2 \frac{1}{2} π \pi 2 1

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Vishwak Srinivasan by Vishwak Srinivasan , 24, India

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1 solution

Given limit is also equal to L = lim ⁡ x → ∞ 2 x + 2 + s i n ( 2 x ) x ( 2 + s i n ( 2 x ) x ) e sin ⁡ ( x ) \displaystyle L = \lim_{x \to \infty} \dfrac{\frac{2}{x} + 2 + \frac{sin(2x)}{x}}{(2 + \frac{sin(2x)}{x})e^{\sin(x)}}

L = 0 + 2 + 0 ( 2 + 0 ) ( a value between 1 e to e ) ( ∵ lim ⁡ x → ∞ sin ⁡ ( x ) ∈ ( − 1 , 1 ) ) \displaystyle L = \dfrac{0 + 2 + 0}{(2 + 0)(\text{a value between} \frac{1}{e}\text { to } e )} (\because \lim_{x \to \infty} \sin(x) \in (-1,1) )

Hence Limit does not exist .

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