What is the limit as x approaches 0 of (10^x-5^x)/x ?

The limit as x approaches 0 of (10^x-5^x)/x is ln(10)-ln(5)

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limit as x approaches 0 of (10^x-5^x)/x

x \to 0 lim (\frac{1 0 ^{x} - 5 ^{x}}{x})

ln (10) - ln (5)

Decimal Notation

0.69314 \dots

Solution steps

x \to 0 lim (\frac{1 0 ^{x} - 5 ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1 0 ^{x} ln ( 10 ) - 5 ^{x} ln ( 5 )}{1})

= x \to 0 lim (\frac{1 0 ^{x} ln ( 10 ) - 5 ^{x} ln ( 5 )}{1})

Plug in the value

x = 0

= \frac{1 0 ^{0} ln ( 10 ) - 5 ^{0} ln ( 5 )}{1}

Simplify

\frac{1 0 ^{0} ln ( 10 ) - 5 ^{0} ln ( 5 )}{1} : ln (10) - ln (5)

= ln (10) - ln (5)

x \to - \infty lim (5 x - 1)

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

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

x \to 3 + lim (2 x - 1)

x \to 0 + lim (\frac{1}{x} + \frac{1}{∣ x ∣})

What is the limit as x approaches 0 of (10^x-5^x)/x ?
The limit as x approaches 0 of (10^x-5^x)/x is ln(10)-ln(5)