What is the limit as x approaches infinity of (8\sqrt[x]{x^4})/9 ?

The limit as x approaches infinity of (8\sqrt[x]{x^4})/9 is 8/9

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limit as x approaches infinity of (8\sqrt[x]{x^4})/9

x \to \infty lim (\frac{8 x x ^{4}}{9})

\frac{8}{9}

Decimal

0.88888 \dots

Solution steps

x \to \infty lim (\frac{8 x x ^{4}}{9})

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

= \frac{8}{9} \cdot x \to \infty lim (x x^{4})

Apply radical rule:

n a^{m} = a^{\frac{m}{n}},

assuming

a \geq 0

= \frac{8}{9} \cdot x \to \infty lim (x^{\frac{4}{x}})

Apply exponent rules

= \frac{8}{9} \cdot x \to \infty lim (e^{\frac{4}{x} l n (x)})

Apply the Limit Chain Rule:

1

= \frac{8}{9} \cdot 1

Simplify

= \frac{8}{9}

What is the limit as x approaches infinity of (8\sqrt[x]{x^4})/9 ?
The limit as x approaches infinity of (8\sqrt[x]{x^4})/9 is 8/9