What is the limit as x approaches 0 of (7^x-5^x)/x ?

The limit as x approaches 0 of (7^x-5^x)/x is ln(7)-ln(5)

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limit as x approaches 0 of (7^x-5^x)/x

x \to 0 lim (\frac{7 ^{x} - 5 ^{x}}{x})

ln (7) - ln (5)

Decimal

0.33647 \dots

Solution steps

x \to 0 lim (\frac{7 ^{x} - 5 ^{x}}{x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{7 ^{x} ln ( 7 ) - 5 ^{x} ln ( 5 )}{1})

Plug in the value

x = 0

= \frac{7 ^{0} ln ( 7 ) - 5 ^{0} ln ( 5 )}{1}

Simplify

\frac{7 ^{0} ln ( 7 ) - 5 ^{0} ln ( 5 )}{1} : ln (7) - ln (5)

= ln (7) - ln (5)

What is the limit as x approaches 0 of (7^x-5^x)/x ?
The limit as x approaches 0 of (7^x-5^x)/x is ln(7)-ln(5)