What is the limit as n approaches infinity of (1-2/n)^{-n} ?

The limit as n approaches infinity of (1-2/n)^{-n} is e^2

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limit as n approaches infinity of (1-2/n)^{-n}

n \to \infty lim ((1 - \frac{2}{n})^{- n})

e^{2}

Decimal

7.38905 \dots

Solution steps

n \to \infty lim ((1 - \frac{2}{n})^{- n})

Simplify

(1 - \frac{2}{n})^{- n} : \frac{1}{\frac{( n - 2 ) ^{n}}{n ^{n}}}

= n \to \infty lim (\frac{1}{\frac{( n - 2 ) ^{n}}{n ^{n}}})

Factor out

n

from

n - 2 : n (1 - \frac{2}{n})

= n \to \infty lim \frac{1}{\frac{( n ( 1 - \frac{2}{n} ) ) ^{n}}{n ^{n}}}

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{n \to \infty} ( 1 )}{lim _{n \to \infty} ( \frac{( n ( 1 - \frac{2}{n} ) ) ^{n}}{n ^{n}} )}

n \to \infty lim (1) = 1

n \to \infty lim (\frac{( n ( 1 - \frac{2}{n} ) ) ^{n}}{n ^{n}}) = \frac{1}{e ^{2}}

= \frac{1}{\frac{1}{e ^{2}}}

Simplify

= e^{2}

x \to 2 - lim (2 x - 5)

x \to 0 lim (x + 4 - 2)

x \to 0 - lim (\frac{x + 5}{x})

x \to 4 lim (3 3 - x)

x \to 0 lim (\frac{1}{x + 4} - \frac{1}{4})

What is the limit as n approaches infinity of (1-2/n)^{-n} ?
The limit as n approaches infinity of (1-2/n)^{-n} is e^2