What is the limit as x approaches pi/4 of tan(2x)tan(pi/4-x) ?

The limit as x approaches pi/4 of tan(2x)tan(pi/4-x) is 1/2

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limit as x approaches pi/4 of tan(2x)tan(pi/4-x)

x \to \frac{π}{4} lim (tan (2 x) tan (\frac{π}{4} - x))

\frac{1}{2}

Decimal Notation

0.5

Solution steps

x \to \frac{π}{4} lim (tan (2 x) tan (\frac{π}{4} - x))

Rewrite for L'Hopital

= x \to \frac{π}{4} lim (\frac{tan ( \frac{π}{4} - x )}{\frac{1}{t a n ( 2 x )}})

Apply L'Hopital's Rule

: x \to \frac{π}{4} lim (\frac{- sec ^{2} ( \frac{π}{4} - x )}{- csc ^{2} ( 2 x ) \cdot 2})

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

Plug in the value

x = \frac{π}{4}

= \frac{- sec ^{2} ( \frac{π}{4} - \frac{π}{4} )}{- csc ^{2} ( 2 \cdot \frac{π}{4} ) \cdot 2}

Simplify

\frac{- sec ^{2} ( \frac{π}{4} - \frac{π}{4} )}{- csc ^{2} ( 2 \cdot \frac{π}{4} ) \cdot 2} : \frac{1}{2}

= \frac{1}{2}

x \to 0 lim ((1 - 7 x)^{\frac{8}{x}})

x \to \infty lim (\frac{π}{2})

x \to - 2 lim (x - 3)

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

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

What is the limit as x approaches pi/4 of tan(2x)tan(pi/4-x) ?
The limit as x approaches pi/4 of tan(2x)tan(pi/4-x) is 1/2