What is the limit as x approaches 0 of (-x-ln(1-x))/(8x^2) ?

The limit as x approaches 0 of (-x-ln(1-x))/(8x^2) is 1/16

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limit as x approaches 0 of (-x-ln(1-x))/(8x^2)

x \to 0 lim (\frac{- x - ln ( 1 - x )}{8 x ^{2}})

\frac{1}{16}

Decimal

0.0625

Solution steps

x \to 0 lim (\frac{- x - ln ( 1 - x )}{8 x ^{2}})

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

= \frac{1}{8} \cdot x \to 0 lim (\frac{- x - ln ( 1 - x )}{x ^{2}})

Apply L'Hopital's Rule

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

= \frac{1}{8} \cdot x \to 0 lim (\frac{- 1 + \frac{1}{1 - x}}{2 x})

Simplify

\frac{- 1 + \frac{1}{1 - x}}{2 x} : \frac{1}{2 ( - x + 1 )}

= \frac{1}{8} \cdot x \to 0 lim (\frac{1}{2 ( - x + 1 )})

Plug in the value

x = 0

= \frac{1}{8} \cdot \frac{1}{2 ( - 0 + 1 )}

Simplify

\frac{1}{8} \cdot \frac{1}{2 ( - 0 + 1 )} : \frac{1}{16}

= \frac{1}{16}

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

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

x \to 0 lim (\frac{csc ^{2} ( x )}{cot ^{2} ( x )})

x \to \infty lim (\frac{6 x ^{2} - 7 x + 10}{5 x ^{3} + 2 x ^{2} - 4 x + 5})

x \to 0 lim (x^{2} lo g_{e} (x))

What is the limit as x approaches 0 of (-x-ln(1-x))/(8x^2) ?
The limit as x approaches 0 of (-x-ln(1-x))/(8x^2) is 1/16