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

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

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

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

ln (5) - ln (10)

Decimal

- 0.69314 \dots

Solution steps

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= ln (5) - ln (10)

x \to 5 lim (x^{3} + 2 x)

\int \frac{1}{2 ( x ^{3} + x ^{2} )} d x

y^{\frac{1}{2}} \frac{d y}{d x} + y^{\frac{3}{2}} = 1, y (0) = 9

17 e^{x} - e x^{e}

\frac{\partial}{\partial x} (e^{- x} sin (yz))

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