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

The limit as x approaches 0 of (2x^2)/(3-3cos(x)) is 4/3

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

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

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

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

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

2 \cdot \frac{2}{3 cos ( 0 )} : \frac{4}{3}

= \frac{4}{3}

x \to 1 lim (\frac{1}{5 - x})

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

x \to 4 lim (\frac{x - 4}{x + 1})

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

x \to 2 lim (\frac{2 x - 5}{x - 2})

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