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

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

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limit as x approaches 0 of ((2^x-5^x))/x

x \to 0 lim (\frac{( 2 ^{x} - 5 ^{x} )}{x})

ln (2) - ln (5)

Decimal

- 0.91629 \dots

Solution steps

x \to 0 lim (\frac{2 ^{x} - 5 ^{x}}{x})

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

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

= ln (2) - ln (5)

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