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

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

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

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

- \infty

Solution steps

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

Factor out

x

from

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

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

Cancel the common factor:

x

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

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

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

x \to 0 + lim (1) = 1

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

= 1 - \infty

Apply Infinity Property:

c - \infty = - \infty

= - \infty

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

x \to 0 lim (\frac{2}{1 + x})

x \to π lim (x sin (x))

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

x \to 3 lim (\frac{x ^{2} + 3 x + 3}{x})

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