What is the limit as x approaches 0 of (9^x-13^x)/x ?

The limit as x approaches 0 of (9^x-13^x)/x is 2ln(3)-ln(13)

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limit as x approaches 0 of (9^x-13^x)/x

x \to 0 lim (\frac{9 ^{x} - 1 3 ^{x}}{x})

2 ln (3) - ln (13)

Decimal Notation

- 0.36772 \dots

Solution steps

x \to 0 lim (\frac{9 ^{x} - 1 3 ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 ln ( 3 ) \cdot 9 ^{x} - 1 3 ^{x} ln ( 13 )}{1})

= x \to 0 lim (\frac{2 ln ( 3 ) \cdot 9 ^{x} - 1 3 ^{x} ln ( 13 )}{1})

Plug in the value

x = 0

= \frac{2 ln ( 3 ) \cdot 9 ^{0} - 1 3 ^{0} ln ( 13 )}{1}

Simplify

\frac{2 ln ( 3 ) \cdot 9 ^{0} - 1 3 ^{0} ln ( 13 )}{1} : 2 ln (3) - ln (13)

= 2 ln (3) - ln (13)

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What is the limit as x approaches 0 of (9^x-13^x)/x ?
The limit as x approaches 0 of (9^x-13^x)/x is 2ln(3)-ln(13)