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)

y = 2 x^{2}, (- 2, 8)

\int_{0}^{5} (t^{2} + 2 t - 15) d t

t \to 0 lim (\frac{5}{t} - \frac{5}{e ^{t} - 1})

y = x^{2} - 1, x = - 2, x = 1

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)