What is the limit as x approaches 0 of ((a^x-b^x))/x ?

The limit as x approaches 0 of ((a^x-b^x))/x is ln(a)-ln(b)

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limit as x approaches 0 of ((a^x-b^x))/x

x \to 0 lim (\frac{( a ^{x} - b ^{x} )}{x})

ln (a) - ln (b)

Solution steps

x \to 0 lim (\frac{a ^{x} - b ^{x}}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{a ^{x} ln ( a ) - b ^{x} ln ( b )}{1})

= x \to 0 lim (\frac{a ^{x} ln ( a ) - b ^{x} ln ( b )}{1})

Plug in the value

x = 0

= \frac{a ^{0} ln ( a ) - b ^{0} ln ( b )}{1}

Simplify

\frac{a ^{0} ln ( a ) - b ^{0} ln ( b )}{1} : ln (a) - ln (b)

= ln (a) - ln (b)

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