What is the limit as x approaches 0 of (1-cos(2x))/(xsin(2x)) ?

The limit as x approaches 0 of (1-cos(2x))/(xsin(2x)) is 1

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limit as x approaches 0 of (1-cos(2x))/(xsin(2x))

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

1

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{4 cos ( 2 x )}{4 cos ( 2 x ) - 4 x sin ( 2 x )})

= x \to 0 lim (\frac{4 cos ( 2 x )}{4 cos ( 2 x ) - 4 x sin ( 2 x )})

Plug in the value

x = 0

= \frac{4 cos ( 2 \cdot 0 )}{4 cos ( 2 \cdot 0 ) - 4 \cdot 0 \cdot sin ( 2 \cdot 0 )}

Simplify

\frac{4 cos ( 2 \cdot 0 )}{4 cos ( 2 \cdot 0 ) - 4 \cdot 0 \cdot sin ( 2 \cdot 0 )} : 1

= 1

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

x \to 4 lim (2 + 4 x)

x \to 3 lim (\frac{2 x + 1}{x - 1})

x \to \frac{π}{6} lim (2 sin (x))

x \to 2 lim (3 x^{3} + 7 x - 16)

What is the limit as x approaches 0 of (1-cos(2x))/(xsin(2x)) ?
The limit as x approaches 0 of (1-cos(2x))/(xsin(2x)) is 1