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

The limit as x approaches 0 of (1-e^x)/(3x) is -1/3

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

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

- \frac{1}{3}

Decimal

- 0.33333 \dots

Solution steps

x \to 0 lim (\frac{1 - e ^{x}}{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{1 - e ^{x}}{x})

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

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

= - \frac{1}{3}

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