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

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

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

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

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

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

Apply L'Hopital's Rule

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

= \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}

f (x) = (4 x - 7)^{\frac{1}{2}}, at x = 4

\int e^{x} x^{2} d x

\frac{1}{e ^{y}} y^{^{'}} = \frac{1}{e ^{x}}

\int \frac{x ^{3} - 8}{x - 2} d x

\frac{5}{12} (3 + 5) a^{3}

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