The limit of the perimeter of the regular n-g | vedclass

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(A) regular $n$-gon inscribed in a circle of radius $R$ has a perimeter given by $P_n = 2nR \sin(\frac{\pi}{n})$.As $n 	o \infty$,we use the limit $\

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$2\pi R$

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(A) regular $n$-gon inscribed in a circle of radius $R$ has a perimeter given by $P_n = 2nR \sin(\frac{\pi}{n})$.As $n 	o \infty$,we use the limit $\lim_{x 	o 0} \frac{\sin x}{x} = 1$.Substituting $x = \frac{\pi}{n}$,as $n 	o \infty$,$x 	o 0$.Thus,$\lim_{n 	o \infty} P_n = \lim_{n 	o \infty} 2nR \sin(\frac{\pi}{n}) = \lim_{n 	o \infty} 2\pi R \frac{\sin(\pi/n)}{\pi/n} = 2\pi R(1) = 2\pi R$.Therefore,the perimeter approaches the circumference of the circle,which is $2\pi R$.

The limit of the perimeter of the regular $n$-gons inscribed in a circle of radius $R$ as $n \to \infty$ is

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