What is the limit as x approaches 0 of ((9^x-5^x))/x ?

The limit as x approaches 0 of ((9^x-5^x))/x is 2ln(3)-ln(5)

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limit as x approaches 0 of ((9^x-5^x))/x

x \to 0 lim (\frac{( 9 ^{x} - 5 ^{x} )}{x})

2 ln (3) - ln (5)

Decimal

0.58778 \dots

Solution steps

x \to 0 lim (\frac{9 ^{x} - 5 ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 ln ( 3 ) \cdot 9 ^{x} - 5 ^{x} ln ( 5 )}{1})

= x \to 0 lim (\frac{2 ln ( 3 ) \cdot 9 ^{x} - 5 ^{x} ln ( 5 )}{1})

Plug in the value

x = 0

= \frac{2 ln ( 3 ) \cdot 9 ^{0} - 5 ^{0} ln ( 5 )}{1}

Simplify

\frac{2 ln ( 3 ) \cdot 9 ^{0} - 5 ^{0} ln ( 5 )}{1} : 2 ln (3) - ln (5)

= 2 ln (3) - ln (5)

y^{^{'}} = y \cdot k (100 - y)

\frac{d}{d x} (- \frac{x ^{6}}{1 + x ^{12}})

\int \frac{x}{( x + 3 ) ^{2}} d x

cot (x) y^{^{'}} = y

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

What is the limit as x approaches 0 of ((9^x-5^x))/x ?
The limit as x approaches 0 of ((9^x-5^x))/x is 2ln(3)-ln(5)