What is the limit as x approaches 0 of (4^x-1)/(5x) ?

The limit as x approaches 0 of (4^x-1)/(5x) is 2/5 ln(2)

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limit as x approaches 0 of (4^x-1)/(5x)

x \to 0 lim (\frac{4 ^{x} - 1}{5 x})

\frac{2}{5} ln (2)

Decimal

0.27725 \dots

Solution steps

x \to 0 lim (\frac{4 ^{x} - 1}{5 x})

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

= \frac{1}{5} \cdot x \to 0 lim (\frac{4 ^{x} - 1}{x})

Apply L'Hopital's Rule

= \frac{1}{5} \cdot x \to 0 lim (\frac{ln ( 2 ) \cdot 2 ^{2 x + 1}}{1})

Plug in the value

x = 0

= \frac{1}{5} \cdot \frac{ln ( 2 ) \cdot 2 ^{2 \cdot 0 + 1}}{1}

Simplify

\frac{1}{5} \cdot \frac{ln ( 2 ) \cdot 2 ^{2 \cdot 0 + 1}}{1} : \frac{2}{5} ln (2)

= \frac{2}{5} ln (2)

What is the limit as x approaches 0 of (4^x-1)/(5x) ?
The limit as x approaches 0 of (4^x-1)/(5x) is 2/5 ln(2)