What is the limit as x approaches 2-of (x-2)*ln(2-x) ?

The limit as x approaches 2-of (x-2)*ln(2-x) is 0

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limit as x approaches 2-of (x-2)*ln(2-x)

x \to 2 - lim ((x - 2) \cdot ln (2 - x))

0

Solution steps

x \to 2 - lim ((x - 2) ln (2 - x))

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

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

= x \to 2 - lim (- x + 2)

Plug in the value

x = 2

= - 2 + 2

Simplify

= 0

x \to 0 lim (\frac{x ^{3} - 3 x}{x})

x \to 0 + lim ((e^{2 x} + x)^{\frac{1}{( 3 x )}})

x \to 0 lim (x ln (3 x))

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

x \to \infty lim (3, x)

What is the limit as x approaches 2-of (x-2)*ln(2-x) ?
The limit as x approaches 2-of (x-2)*ln(2-x) is 0