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

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

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

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

- \infty

Solution steps

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

\frac{e ^{2 x}}{x} = e^{2 x} \frac{1}{x}

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

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

With the exception of indeterminate form

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

x \to 0 - lim (e^{2 x}) = 1

x \to 0 - lim (\frac{1}{x}) = - \infty

= \frac{1}{3} \cdot 1 \cdot (- \infty)

Apply Infinity Property:

c \cdot (- \infty) = - \infty

= - \infty

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