What is the limit as x approaches infinity of (((4+x))/x)^{3+x/3} ?

The limit as x approaches infinity of (((4+x))/x)^{3+x/3} is e^{4/3}

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limit as x approaches infinity of (((4+x))/x)^{3+x/3}

x \to \infty lim ((\frac{( 4 + x )}{x})^{3 + \frac{x}{3}})

e^{\frac{4}{3}}

Solution steps

x \to \infty lim ((\frac{4 + x}{x})^{3 + \frac{x}{3}})

Simplify

(\frac{4 + x}{x})^{3 + \frac{x}{3}} : (\frac{4 + x}{x})^{3} (\frac{4 + x}{x})^{\frac{x}{3}}

= x \to \infty lim ((\frac{4 + x}{x})^{3} (\frac{4 + x}{x})^{\frac{x}{3}})

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{4 + x}{x})^{3}) \cdot x \to \infty lim ((\frac{4 + x}{x})^{\frac{x}{3}})

x \to \infty lim ((\frac{4 + x}{x})^{3}) = 1

x \to \infty lim ((\frac{4 + x}{x})^{\frac{x}{3}}) = e^{\frac{4}{3}}

= 1 \cdot e^{\frac{4}{3}}

Simplify

= e^{\frac{4}{3}}

x^{2} - 1 x y + x \frac{d y}{d x} = 0

\int ln ∣ x - 2 ∣ d x

y^{^{'}} = y x, y (1) = 7

\int_{4}^{9} (3 x) d x

y = x^{2} + 2

What is the limit as x approaches infinity of (((4+x))/x)^{3+x/3} ?
The limit as x approaches infinity of (((4+x))/x)^{3+x/3} is e^{4/3}