What is the limit as x approaches 0+of (1+ln(x))/x ?

The limit as x approaches 0+of (1+ln(x))/x is -infinity

limit as x approaches 0+of (1+ln(x))/x

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

- \infty

Solution steps

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

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

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

x \to 0 + lim (1 + ln (x)) = - \infty

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

= (- \infty) \cdot \infty

Apply Infinity Property:

(- \infty) \infty = - \infty

= - \infty

x \to \infty lim (x^{2} - 5 x)

x \to 5 lim (2 x - 1)

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

x \to 0 lim (1 - x)

x \to \infty lim (ln (3 x - 7))

What is the limit as x approaches 0+of (1+ln(x))/x ?
The limit as x approaches 0+of (1+ln(x))/x is -infinity