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

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

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limit as x approaches 0 of ((9^x-3^x))/x

x \to 0 lim (\frac{( 9 ^{x} - 3 ^{x} )}{x})

ln (3)

Decimal Notation

1.09861 \dots

Solution steps

x \to 0 lim (\frac{9 ^{x} - 3 ^{x}}{x})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= ln (3)

\int x \cdot ln^{3} (x) d x

x \to 2 lim (2 x^{2} - 3 x + 5)

\frac{s ^{2}}{( s ^{2} + \frac{1}{4} )}

\int \frac{( x + 1 ) ^{2}}{x} d x

sin (3 x)

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