What is the limit as x approaches 0 of (3x^2)/(1-cos^2(x/2)) ?

The limit as x approaches 0 of (3x^2)/(1-cos^2(x/2)) is 12

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limit as x approaches 0 of (3x^2)/(1-cos^2(x/2))

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

12

Solution steps

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

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

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 x}{\frac{s i n ( x )}{2}})

= 3 \cdot x \to 0 lim (\frac{2 x}{\frac{s i n ( x )}{2}})

Simplify

\frac{2 x}{\frac{s i n ( x )}{2}} : 4 x csc (x)

= 3 \cdot x \to 0 lim (4 x csc (x))

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

= 3 \cdot 4 \cdot x \to 0 lim (x csc (x))

Rewrite for L'Hopital

= 3 \cdot 4 \cdot x \to 0 lim (\frac{x}{\frac{1}{c s c ( x )}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( x )})

= 3 \cdot 4 \cdot x \to 0 lim (\frac{1}{cos ( x )})

Plug in the value

x = 0

= 3 \cdot 4 \cdot \frac{1}{cos ( 0 )}

Simplify

3 \cdot 4 \cdot \frac{1}{cos ( 0 )} : 12

= 12

\int \frac{x ^{3} + x ^{2} + 2}{x ^{2} - x} d x

\int \frac{1}{( x ^{2} + 2 x + 3 ) ^{\frac{3}{2}}} d x

\frac{d y}{d x} = 6 y x

x \to 0 lim (\frac{5}{x})

\frac{d y}{d x} + 5 e^{x + y} = 0

What is the limit as x approaches 0 of (3x^2)/(1-cos^2(x/2)) ?
The limit as x approaches 0 of (3x^2)/(1-cos^2(x/2)) is 12