What is the limit as x approaches 0 of (sin^2(3x))/(4x^2) ?

The limit as x approaches 0 of (sin^2(3x))/(4x^2) is 9/4

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limit as x approaches 0 of (sin^2(3x))/(4x^2)

x \to 0 lim (\frac{sin ^{2} ( 3 x )}{4 x ^{2}})

\frac{9}{4}

Decimal

2.25

Solution steps

x \to 0 lim (\frac{sin ^{2} ( 3 x )}{4 x ^{2}})

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

= \frac{1}{4} \cdot x \to 0 lim (\frac{sin ^{2} ( 3 x )}{x ^{2}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{3 sin ( 6 x )}{2 x})

= \frac{1}{4} \cdot x \to 0 lim (\frac{3 sin ( 6 x )}{2 x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{18 cos ( 6 x )}{2})

= \frac{1}{4} \cdot x \to 0 lim (\frac{18 cos ( 6 x )}{2})

Plug in the value

x = 0

= \frac{1}{4} \cdot \frac{18 cos ( 6 \cdot 0 )}{2}

Simplify

\frac{1}{4} \cdot \frac{18 cos ( 6 \cdot 0 )}{2} : \frac{9}{4}

= \frac{9}{4}

x \to 0 lim (4 x + 5)

x \to 0 lim (+ (x^{2} ln (x)))

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

x \to 0 lim (+ x^{x})

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

What is the limit as x approaches 0 of (sin^2(3x))/(4x^2) ?
The limit as x approaches 0 of (sin^2(3x))/(4x^2) is 9/4