límite cuando x tiende a 1+de (1-x)*tan((pix)/2)

Solución

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

\frac{2}{π}

Notación decimal

0.63661 \dots

Pasos de solución

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

Aplicar la siguiente propiedad algebraica

: a \cdot b = \frac{a}{\frac{1}{b}}, b \neq = 0

(1 - x) tan (\frac{x π}{2}) = \frac{tan ( \frac{x π}{2} )}{\frac{1}{( 1 - x )}}

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

Expandir

\frac{tan ( \frac{x π}{2} )}{\frac{1}{1 - x}} : tan (\frac{π x}{2}) - x tan (\frac{π x}{2})

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

Multiplicar por el conjugado de

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

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

Simplificar

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

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

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

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

Reescribir para ajustar por L’Hopital

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

Aplicar la regla de L'Hopital

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

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

Sustituir la variable

= - \frac{1}{- csc ^{2} ( \frac{π 1}{2} ) \frac{π}{2}}

Simplificar

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

= \frac{2}{π}

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