What is the limit as x approaches 0 of ((5^x-4^x))/x ?

The limit as x approaches 0 of ((5^x-4^x))/x is ln(5)-2ln(2)

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limit as x approaches 0 of ((5^x-4^x))/x

x \to 0 lim (\frac{( 5 ^{x} - 4 ^{x} )}{x})

ln (5) - 2 ln (2)

Decimal Notation

0.22314 \dots

Solution steps

x \to 0 lim (\frac{5 ^{x} - 4 ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{5 ^{x} ln ( 5 ) - ln ( 2 ) \cdot 2 ^{2 x + 1}}{1})

= x \to 0 lim (\frac{5 ^{x} ln ( 5 ) - ln ( 2 ) \cdot 2 ^{2 x + 1}}{1})

Plug in the value

x = 0

= \frac{5 ^{0} ln ( 5 ) - ln ( 2 ) \cdot 2 ^{2 \cdot 0 + 1}}{1}

Simplify

\frac{5 ^{0} ln ( 5 ) - ln ( 2 ) \cdot 2 ^{2 \cdot 0 + 1}}{1} : ln (5) - 2 ln (2)

= ln (5) - 2 ln (2)

x \to 0 lim (cos (\frac{2 π}{x}))

x \to 0 lim (\frac{3 x}{∣ x ∣})

x \to 0 + lim (x^{- x})

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

x \to 3 lim (5 - 2 x)

What is the limit as x approaches 0 of ((5^x-4^x))/x ?
The limit as x approaches 0 of ((5^x-4^x))/x is ln(5)-2ln(2)