What is the limit as x approaches infinity of ((x+3)/x)^{2x-7} ?

The limit as x approaches infinity of ((x+3)/x)^{2x-7} is e^6

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limit as x approaches infinity of ((x+3)/x)^{2x-7}

x \to \infty lim ((\frac{x + 3}{x})^{2 x - 7})

e^{6}

Decimal

403.42879 \dots

Solution steps

x \to \infty lim ((\frac{x + 3}{x})^{2 x - 7})

Simplify

(\frac{x + 3}{x})^{2 x - 7} : (\frac{x + 3}{x})^{2 x} (\frac{x + 3}{x})^{- 7}

= x \to \infty lim ((\frac{x + 3}{x})^{2 x} (\frac{x + 3}{x})^{- 7})

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 ((\frac{x + 3}{x})^{2 x}) \cdot x \to \infty lim ((\frac{x + 3}{x})^{- 7})

x \to \infty lim ((\frac{x + 3}{x})^{2 x}) = e^{6}

x \to \infty lim ((\frac{x + 3}{x})^{- 7}) = 1

= e^{6} \cdot 1

Simplify

= e^{6}

What is the limit as x approaches infinity of ((x+3)/x)^{2x-7} ?
The limit as x approaches infinity of ((x+3)/x)^{2x-7} is e^6