What is the limit as x approaches 0 of ((9^x-8^x))/x ?

The limit as x approaches 0 of ((9^x-8^x))/x is 2ln(3)-3ln(2)

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limit as x approaches 0 of ((9^x-8^x))/x

x \to 0 lim (\frac{( 9 ^{x} - 8 ^{x} )}{x})

2 ln (3) - 3 ln (2)

Decimal

0.11778 \dots

Solution steps

x \to 0 lim (\frac{9 ^{x} - 8 ^{x}}{x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{2 ln ( 3 ) \cdot 9 ^{x} - 3 ln ( 2 ) \cdot 8 ^{x}}{1})

Plug in the value

x = 0

= \frac{2 ln ( 3 ) \cdot 9 ^{0} - 3 ln ( 2 ) \cdot 8 ^{0}}{1}

Simplify

\frac{2 ln ( 3 ) \cdot 9 ^{0} - 3 ln ( 2 ) \cdot 8 ^{0}}{1} : 2 ln (3) - 3 ln (2)

= 2 ln (3) - 3 ln (2)

What is the limit as x approaches 0 of ((9^x-8^x))/x ?
The limit as x approaches 0 of ((9^x-8^x))/x is 2ln(3)-3ln(2)