Explained: "Limit of limn→+∞(cos⁡xn+sin⁡2xn)n"

avalg10o

Answered question

2022-03-22

Limit of

lim_{n \to + \infty} {(\cos \frac{x}{n} + \sin \frac{2 x}{n})}^{n}

Karsyn Wu

Beginner2022-03-23Added 17 answers

By using the following Taylor expansions at 0,

\cos (t) = 1 + o (t)

\sin (t) = t + o (t)

, and

\ln (1 + t) = t + o (t)

, we have that

{(\cos (\frac{x}{n}) + \sin (\frac{2 x}{n}))}^{n} = \exp (n \ln (\cos (\frac{x}{n}) + \sin (\frac{2 x}{n})))

= \exp (n \ln (1 + \frac{2 x}{n} + o (\frac{1}{n})))

= \exp (n (\frac{2 x}{n} + o (\frac{1}{n})))

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