What is the limit as x approaches infinity of (1+sin(3/x))^{x+1} ?

The limit as x approaches infinity of (1+sin(3/x))^{x+1} is e^3

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limit as x approaches infinity of (1+sin(3/x))^{x+1}

x \to \infty lim ((1 + sin (\frac{3}{x}))^{x + 1})

e^{3}

Decimal

20.08553 \dots

Solution steps

x \to \infty lim ((1 + sin (\frac{3}{x}))^{x + 1})

Simplify

(1 + sin (\frac{3}{x}))^{x + 1} : (1 + sin (\frac{3}{x}))^{x} (1 + sin (\frac{3}{x}))

= x \to \infty lim ((1 + sin (\frac{3}{x}))^{x} (1 + sin (\frac{3}{x})))

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

= x \to \infty lim ((1 + sin (\frac{3}{x}))^{x}) \cdot x \to \infty lim (1 + sin (\frac{3}{x}))

x \to \infty lim ((1 + sin (\frac{3}{x}))^{x}) = e^{3}

x \to \infty lim (1 + sin (\frac{3}{x})) = 1

= e^{3} \cdot 1

Simplify

= e^{3}

x \to 0 lim (\frac{x}{x ^{3}})

x \to 0 lim (\frac{2 ^{x} - 1}{x ^{2}})

x \to 3 lim (\frac{2 x - 3}{x - 3})

x \to 1 lim (x^{\frac{1}{( x - 1 )}})

x \to - \infty lim (x, 0)

What is the limit as x approaches infinity of (1+sin(3/x))^{x+1} ?
The limit as x approaches infinity of (1+sin(3/x))^{x+1} is e^3