What is the limit as x approaches 1 of ((1+x)(1-x)}{x*\frac{sin(pix))/x} ?

The limit as x approaches 1 of ((1+x)(1-x)}{x*\frac{sin(pix))/x} is 2/pi

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limit as x approaches 1 of ((1+x)(1-x)}{x*\frac{sin(pix))/x}

x \to 1 lim (\frac{( 1 + x ) ( 1 - x )}{x \cdot \frac{s i n ( π x )}{x}})

\frac{2}{π}

Decimal Notation

0.63661 \dots

Solution steps

x \to 1 lim (\frac{( 1 + x ) ( 1 - x )}{x \frac{s i n ( π x )}{x}})

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

\frac{( 1 + x ) ( 1 - x )}{x \frac{s i n ( π x )}{x}} = \frac{( x ( \frac{1}{x} + 1 ) ) ( x ( \frac{1}{x} - 1 ) )}{x \frac{s i n ( π x )}{x}}

= x \to 1 lim (\frac{x ( \frac{1}{x} + 1 ) x ( \frac{1}{x} - 1 )}{x \frac{s i n ( π x )}{x}})

Simplify

\frac{( x ( \frac{1}{x} + 1 ) ) ( x ( \frac{1}{x} - 1 ) )}{x \frac{s i n ( π x )}{x}} : \frac{1 - x ^{2}}{sin ( π x )}

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 1

= \frac{- 2 \cdot 1}{cos ( π 1 ) π}

Simplify

\frac{- 2 \cdot 1}{cos ( π 1 ) π} : \frac{2}{π}

= \frac{2}{π}

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What is the limit as x approaches 1 of ((1+x)(1-x)}{x*\frac{sin(pix))/x} ?
The limit as x approaches 1 of ((1+x)(1-x)}{x*\frac{sin(pix))/x} is 2/pi