What is the limit as x approaches 0 of (5^x-6^x)/x ?

The limit as x approaches 0 of (5^x-6^x)/x is ln(5)-ln(6)

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limit as x approaches 0 of (5^x-6^x)/x

x \to 0 lim (\frac{5 ^{x} - 6 ^{x}}{x})

ln (5) - ln (6)

Decimal Notation

- 0.18232 \dots

Solution steps

x \to 0 lim (\frac{5 ^{x} - 6 ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{5 ^{x} ln ( 5 ) - 6 ^{x} ln ( 6 )}{1})

= x \to 0 lim (\frac{5 ^{x} ln ( 5 ) - 6 ^{x} ln ( 6 )}{1})

Plug in the value

x = 0

= \frac{5 ^{0} ln ( 5 ) - 6 ^{0} ln ( 6 )}{1}

Simplify

\frac{5 ^{0} ln ( 5 ) - 6 ^{0} ln ( 6 )}{1} : ln (5) - ln (6)

= ln (5) - ln (6)

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

x \to 0 + lim (x ∣ x ∣)

x \to 0 lim (\frac{8}{x})

x \to - \infty lim (2 (x))

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

What is the limit as x approaches 0 of (5^x-6^x)/x ?
The limit as x approaches 0 of (5^x-6^x)/x is ln(5)-ln(6)