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

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

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

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

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

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

Simplify

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

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

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

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

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

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

= \frac{1}{e} \cdot 1

Simplify

= \frac{1}{e}

x \to - \infty lim (x - x^{2})

x \to 20 lim (x^{2} + 1)

h \to 0 lim (\frac{1}{h + 1})

x \to 4 + lim (x^{2} + 7 - x)

x \to 6 lim (8 6 + x)

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