What is the limit as x approaches 0 of ((9^x-4^x))/x ?

The limit as x approaches 0 of ((9^x-4^x))/x is 2ln(3)-2ln(2)

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limit as x approaches 0 of ((9^x-4^x))/x

x \to 0 lim (\frac{( 9 ^{x} - 4 ^{x} )}{x})

2 ln (3) - 2 ln (2)

Decimal

0.81093 \dots

Solution steps

x \to 0 lim (\frac{9 ^{x} - 4 ^{x}}{x})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

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

x \to 2 lim (4 (x - 4) (x - 2))

x \to \infty lim (x - 7)

x \to 0 lim (\frac{a}{x ^{2}})

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

x \to \infty lim (x, 2)

What is the limit as x approaches 0 of ((9^x-4^x))/x ?
The limit as x approaches 0 of ((9^x-4^x))/x is 2ln(3)-2ln(2)