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

The limit as x approaches 0 of ((1-cos(x)))/(3x^2) is 1/6

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

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

\frac{1}{6}

Decimal Notation

0.16666 \dots

Solution steps

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

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

\frac{1}{3} \cdot \frac{cos ( 0 )}{2} : \frac{1}{6}

= \frac{1}{6}

x \to 4 lim (\frac{4 - 4 + 3 x}{4 - x})

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

x \to 25 lim (5 - x)

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

x \to \frac{1}{8} lim (\frac{1}{x} + x - 1)

What is the limit as x approaches 0 of ((1-cos(x)))/(3x^2) ?
The limit as x approaches 0 of ((1-cos(x)))/(3x^2) is 1/6