What is the limit as x approaches 1-of (1-x)ln(1-x) ?

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

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limit as x approaches 1-of (1-x)ln(1-x)

x \to 1 - lim ((1 - x) ln (1 - x))

0

Solution steps

x \to 1 - lim ((1 - x) ln (1 - x))

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

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

= x \to 1 - lim (- 1 + x)

Plug in the value

x = 1

= - 1 + 1

Simplify

= 0

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

x \to 0 lim ((1 - x)^{x})

x \to \frac{2}{π} lim (e^{s e c (x)})

x \to 0 - lim (ln (tan (x)))

x \to 2 lim (\frac{10}{x ^{2} - 4})

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