What is the limit as x approaches infinity of (sqrt(x))/(ln(x)) ?

The limit as x approaches infinity of (sqrt(x))/(ln(x)) is infinity

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limit as x approaches infinity of (sqrt(x))/(ln(x))

x \to \infty lim (\frac{x}{ln ( x )})

\infty

Solution steps

x \to \infty lim (\frac{x}{ln ( x )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{\frac{1}{2 x}}{\frac{1}{x}})

= x \to \infty lim (\frac{\frac{1}{2 x}}{\frac{1}{x}})

Simplify

\frac{\frac{1}{2 x}}{\frac{1}{x}} : \frac{1}{2} x

= x \to \infty lim (\frac{1}{2} x)

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

= \frac{1}{2} \cdot x \to \infty lim (x)

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

= \frac{1}{2} x \to \infty lim (x)

Apply the common limit:

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

= \frac{1}{2} \infty

Simplify

\frac{1}{2} \infty : \infty

= \infty

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What is the limit as x approaches infinity of (sqrt(x))/(ln(x)) ?
The limit as x approaches infinity of (sqrt(x))/(ln(x)) is infinity