What is the limit as x approaches 0 of (12^x-13^x)/x ?

The limit as x approaches 0 of (12^x-13^x)/x is ln(12)-ln(13)

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limit as x approaches 0 of (12^x-13^x)/x

x \to 0 lim (\frac{1 2 ^{x} - 1 3 ^{x}}{x})

ln (12) - ln (13)

Decimal

- 0.08004 \dots

Solution steps

x \to 0 lim (\frac{1 2 ^{x} - 1 3 ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1 2 ^{x} ln ( 12 ) - 1 3 ^{x} ln ( 13 )}{1})

= x \to 0 lim (\frac{1 2 ^{x} ln ( 12 ) - 1 3 ^{x} ln ( 13 )}{1})

Plug in the value

x = 0

= \frac{1 2 ^{0} ln ( 12 ) - 1 3 ^{0} ln ( 13 )}{1}

Simplify

\frac{1 2 ^{0} ln ( 12 ) - 1 3 ^{0} ln ( 13 )}{1} : ln (12) - ln (13)

= ln (12) - ln (13)

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x \to \infty lim (x \frac{x}{x + 1})

What is the limit as x approaches 0 of (12^x-13^x)/x ?
The limit as x approaches 0 of (12^x-13^x)/x is ln(12)-ln(13)