What is the limit as x approaches pi of (ln(cos(2x)))/((x-pi)^2) ?

The limit as x approaches pi of (ln(cos(2x)))/((x-pi)^2) is -2

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limit as x approaches pi of (ln(cos(2x)))/((x-pi)^2)

x \to π lim (\frac{ln ( cos ( 2 x ) )}{( x - π ) ^{2}})

- 2

Solution steps

x \to π lim (\frac{ln ( cos ( 2 x ) )}{( x - π ) ^{2}})

Apply L'Hopital's Rule

= x \to π lim (\frac{- 2 tan ( 2 x )}{2 ( x - π )})

Apply L'Hopital's Rule

= x \to π lim (\frac{- 4 sec ^{2} ( 2 x )}{2})

Plug in the value

x = π

= \frac{- 4 sec ^{2} ( 2 π )}{2}

Simplify

\frac{- 4 sec ^{2} ( 2 π )}{2} : - 2

= - 2

What is the limit as x approaches pi of (ln(cos(2x)))/((x-pi)^2) ?
The limit as x approaches pi of (ln(cos(2x)))/((x-pi)^2) is -2