What is the limit as x approaches 0 of (a^x-1)/(2x) ?

The limit as x approaches 0 of (a^x-1)/(2x) is 1/2 ln(a)

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limit as x approaches 0 of (a^x-1)/(2x)

x \to 0 lim (\frac{a ^{x} - 1}{2 x})

\frac{1}{2} ln (a)

Solution steps

x \to 0 lim (\frac{a ^{x} - 1}{2 x})

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

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

Apply L'Hopital's Rule

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

= \frac{1}{2} \cdot x \to 0 lim (\frac{a ^{x} ln ( a )}{1})

Plug in the value

x = 0

= \frac{1}{2} \cdot \frac{a ^{0} ln ( a )}{1}

Simplify

\frac{1}{2} \cdot \frac{a ^{0} ln ( a )}{1} : \frac{1}{2} ln (a)

= \frac{1}{2} ln (a)

x \to - \infty lim (4 x^{3})

x \to 0 lim (\frac{8}{x + 2})

\frac{\partial}{\partial x} (x^{2} + y^{2} + 2)

\frac{1}{b - m t}

n = 5 \sum \infty cos (\frac{1}{n ^{3}})

What is the limit as x approaches 0 of (a^x-1)/(2x) ?
The limit as x approaches 0 of (a^x-1)/(2x) is 1/2 ln(a)