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

The limit as x approaches-infinity of x+3x^{2/3} is -infinity

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

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

- \infty

Solution steps

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

Factor out

x

from

x + 3 x^{\frac{2}{3}} : x (1 + \frac{3 x ^{\frac{2}{3}}}{x})

= x \to - \infty lim (x (1 + \frac{3 x ^{\frac{2}{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 (x) \cdot x \to - \infty lim (1 + \frac{3 x ^{\frac{2}{3}}}{x})

x \to - \infty lim (x) = - \infty

x \to - \infty lim (1 + \frac{3 x ^{\frac{2}{3}}}{x}) = 1

= (- \infty) \cdot 1

Apply Infinity Property:

c \cdot (- \infty) = - \infty

= - \infty

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