What is the limit as x approaches 0+of (ln(5x))/x ?

The limit as x approaches 0+of (ln(5x))/x is -infinity

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limit as x approaches 0+of (ln(5x))/x

x \to 0 + lim (\frac{ln ( 5 x )}{x})

- \infty

Solution steps

x \to 0 + lim (\frac{ln ( 5 x )}{x})

\frac{ln ( 5 x )}{x} = ln (5 x) \frac{1}{x}

= x \to 0 + lim (ln (5 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 0 + lim (ln (5 x)) \cdot x \to 0 + lim (\frac{1}{x})

x \to 0 + lim (ln (5 x)) = - \infty

x \to 0 + lim (\frac{1}{x}) = \infty

= (- \infty) \cdot \infty

Apply Infinity Property:

(- \infty) \infty = - \infty

= - \infty

What is the limit as x approaches 0+of (ln(5x))/x ?
The limit as x approaches 0+of (ln(5x))/x is -infinity