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

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

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

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

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

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

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{2 e ^{x} - 2}{x})

Apply L'Hopital's Rule

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

Plug in the value

x = 0

= \frac{1}{3} \cdot \frac{2 e ^{0}}{1}

Simplify

\frac{1}{3} \cdot \frac{2 e ^{0}}{1} : \frac{2}{3}

= \frac{2}{3}

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