What is the limit as x approaches infinity of \sqrt[x]{rx} ?

The limit as x approaches infinity of \sqrt[x]{rx} is 1

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limit as x approaches infinity of \sqrt[x]{rx}

x \to \infty lim (x r x)

1

Solution steps

x \to \infty lim (x r x)

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x r x = (r)^{\frac{1}{x}} (x)^{\frac{1}{x}}

= x \to \infty lim (r^{\frac{1}{x}} x^{\frac{1}{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 (r^{\frac{1}{x}}) \cdot x \to \infty lim (x^{\frac{1}{x}})

x \to \infty lim (r^{\frac{1}{x}}) = 1

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

= 1 \cdot 1

Simplify

= 1

x \to 101 lim (x^{3} - 4 x^{2} + 7)

x \to 0 lim (\frac{2 + x}{x})

x \to 0 lim (3 x + 6)

x \to \infty lim (\frac{x + 7}{7 + 7 ^{x + 1} ( x + 6 )})

x \to - 3 lim (\frac{2 x - 1}{x - 3})

What is the limit as x approaches infinity of \sqrt[x]{rx} ?
The limit as x approaches infinity of \sqrt[x]{rx} is 1