What is the limit as x approaches 0 of (1-cos(3x))/(10x^2) ?

The limit as x approaches 0 of (1-cos(3x))/(10x^2) is 9/20

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limit as x approaches 0 of (1-cos(3x))/(10x^2)

x \to 0 lim (\frac{1 - cos ( 3 x )}{10 x ^{2}})

\frac{9}{20}

Decimal Notation

0.45

Solution steps

x \to 0 lim (\frac{1 - cos ( 3 x )}{10 x ^{2}})

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

= \frac{1}{10} \cdot x \to 0 lim (\frac{1 - cos ( 3 x )}{x ^{2}})

Apply L'Hopital's Rule

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

= \frac{1}{10} \cdot x \to 0 lim (\frac{3 sin ( 3 x )}{2 x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{9 cos ( 3 x )}{2})

= \frac{1}{10} \cdot x \to 0 lim (\frac{9 cos ( 3 x )}{2})

Plug in the value

x = 0

= \frac{1}{10} \cdot \frac{9 cos ( 3 \cdot 0 )}{2}

Simplify

\frac{1}{10} \cdot \frac{9 cos ( 3 \cdot 0 )}{2} : \frac{9}{20}

= \frac{9}{20}

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

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

x \to 5 lim (\frac{∣ x - 5 ∣}{x + 1})

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

x \to 3 - lim (arcsin (\frac{x}{3}))

What is the limit as x approaches 0 of (1-cos(3x))/(10x^2) ?
The limit as x approaches 0 of (1-cos(3x))/(10x^2) is 9/20