What is the limit as x approaches 0 of (8^x-10^x)/x ?

The limit as x approaches 0 of (8^x-10^x)/x is 3ln(2)-ln(10)

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limit as x approaches 0 of (8^x-10^x)/x

x \to 0 lim (\frac{8 ^{x} - 1 0 ^{x}}{x})

3 ln (2) - ln (10)

Decimal

- 0.22314 \dots

Solution steps

x \to 0 lim (\frac{8 ^{x} - 1 0 ^{x}}{x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{3 ln ( 2 ) \cdot 8 ^{x} - 1 0 ^{x} ln ( 10 )}{1})

Plug in the value

x = 0

= \frac{3 ln ( 2 ) \cdot 8 ^{0} - 1 0 ^{0} ln ( 10 )}{1}

Simplify

\frac{3 ln ( 2 ) \cdot 8 ^{0} - 1 0 ^{0} ln ( 10 )}{1} : 3 ln (2) - ln (10)

= 3 ln (2) - ln (10)

What is the limit as x approaches 0 of (8^x-10^x)/x ?
The limit as x approaches 0 of (8^x-10^x)/x is 3ln(2)-ln(10)