What is the limit as x approaches 0 of (3^x-10^x)/x ?

The limit as x approaches 0 of (3^x-10^x)/x is ln(3)-ln(10)

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limit as x approaches 0 of (3^x-10^x)/x

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

ln (3) - ln (10)

Decimal Notation

- 1.20397 \dots

Solution steps

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{3 ^{x} ln ( 3 ) - 1 0 ^{x} ln ( 10 )}{1})

= x \to 0 lim (\frac{3 ^{x} ln ( 3 ) - 1 0 ^{x} ln ( 10 )}{1})

Plug in the value

x = 0

= \frac{3 ^{0} ln ( 3 ) - 1 0 ^{0} ln ( 10 )}{1}

Simplify

\frac{3 ^{0} ln ( 3 ) - 1 0 ^{0} ln ( 10 )}{1} : ln (3) - ln (10)

= ln (3) - ln (10)

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\int_{0}^{1} 8 e^{- 2 x} d x

x \to 8 lim (x - 8)

What is the limit as x approaches 0 of (3^x-10^x)/x ?
The limit as x approaches 0 of (3^x-10^x)/x is ln(3)-ln(10)