What is the limit as x approaches 0 of (13^x-4^x)/x ?

The limit as x approaches 0 of (13^x-4^x)/x is ln(13)-2ln(2)

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limit as x approaches 0 of (13^x-4^x)/x

x \to 0 lim (\frac{1 3 ^{x} - 4 ^{x}}{x})

ln (13) - 2 ln (2)

Decimal

1.17865 \dots

Solution steps

x \to 0 lim (\frac{1 3 ^{x} - 4 ^{x}}{x})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= ln (13) - 2 ln (2)

x \to 0 lim (3 - 7 x + 6 x^{2})

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

x \to - 1 lim (2 x - 15)

x \to \infty lim (3 3 x - 3)

x \to 4 lim (\frac{1}{x} + x^{- 1})

What is the limit as x approaches 0 of (13^x-4^x)/x ?
The limit as x approaches 0 of (13^x-4^x)/x is ln(13)-2ln(2)