What is the limit as x approaches 0 of (arctan(x)-arcsin(x))/(x(1-cos(x))) ?

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limit as x approaches 0 of (arctan(x)-arcsin(x))/(x(1-cos(x)))

x \to 0 lim (\frac{arctan ( x ) - arcsin ( x )}{x ( 1 - cos ( x ) )})

- 1

Solution steps

x \to 0 lim (\frac{arctan ( x ) - arcsin ( x )}{x ( 1 - cos ( x ) )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{\frac{1}{x ^{2} + 1} - \frac{1}{1 - x ^{2}}}{1 - cos ( x ) + x sin ( x )})

= x \to 0 lim (\frac{\frac{1}{x ^{2} + 1} - \frac{1}{1 - x ^{2}}}{1 - cos ( x ) + x sin ( x )})

Apply L'Hopital's Rule

: x \to 0 lim \frac{- \frac{2 x}{( x ^{2} + 1 ) ^{2}} - \frac{x}{( 1 - x ^{2} ) ^{\frac{3}{2}}}}{x cos ( x ) + 2 sin ( x )}

= x \to 0 lim \frac{- \frac{2 x}{( x ^{2} + 1 ) ^{2}} - \frac{x}{( 1 - x ^{2} ) ^{\frac{3}{2}}}}{x cos ( x ) + 2 sin ( x )}

Apply L'Hopital's Rule

: x \to 0 lim \frac{- \frac{2 ( - 3 x ^{2} + 1 )}{( x ^{2} + 1 ) ^{3}} - \frac{2 x ^{2} + 1}{( 1 - x ^{2} ) ^{\frac{5}{2}}}}{- x sin ( x ) + 3 cos ( x )}

= x \to 0 lim \frac{- \frac{2 ( - 3 x ^{2} + 1 )}{( x ^{2} + 1 ) ^{3}} - \frac{2 x ^{2} + 1}{( 1 - x ^{2} ) ^{\frac{5}{2}}}}{- x sin ( x ) + 3 cos ( x )}

Plug in the value

x = 0

= \frac{- \frac{2 ( - 3 \cdot 0 ^{2} + 1 )}{( 0 ^{2} + 1 ) ^{3}} - \frac{2 \cdot 0 ^{2} + 1}{( 1 - 0 ^{2} ) ^{\frac{5}{2}}}}{- 0 \cdot sin ( 0 ) + 3 cos ( 0 )}

Simplify

\frac{- \frac{2 ( - 3 \cdot 0 ^{2} + 1 )}{( 0 ^{2} + 1 ) ^{3}} - \frac{2 \cdot 0 ^{2} + 1}{( 1 - 0 ^{2} ) ^{\frac{5}{2}}}}{- 0 \cdot sin ( 0 ) + 3 cos ( 0 )} : - 1

= - 1

x \to 3 - lim (\frac{3}{( x - 3 ) ^{2}})

x \to 2 lim (\frac{x ^{2} + 5}{x - 3})

x \to 1 + lim (x^{\frac{1}{x ^{4} - 1}})

x \to 4 lim (x^{2} + 3 x - 8)

x \to 0 lim ((1 + x)^{l n (x)})

What is the limit as x approaches 0 of (arctan(x)-arcsin(x))/(x(1-cos(x))) ?
The limit as x approaches 0 of (arctan(x)-arcsin(x))/(x(1-cos(x))) is -1