What is the limit as t approaches 0 of (8^t-5^t)/t ?

The limit as t approaches 0 of (8^t-5^t)/t is 3ln(2)-ln(5)

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limit as t approaches 0 of (8^t-5^t)/t

t \to 0 lim (\frac{8 ^{t} - 5 ^{t}}{t})

3 ln (2) - ln (5)

Decimal

0.47000 \dots

Solution steps

t \to 0 lim (\frac{8 ^{t} - 5 ^{t}}{t})

Apply L'Hopital's Rule

= t \to 0 lim (\frac{3 ln ( 2 ) \cdot 8 ^{t} - 5 ^{t} ln ( 5 )}{1})

Plug in the value

t = 0

= \frac{3 ln ( 2 ) \cdot 8 ^{0} - 5 ^{0} ln ( 5 )}{1}

Simplify

\frac{3 ln ( 2 ) \cdot 8 ^{0} - 5 ^{0} ln ( 5 )}{1} : 3 ln (2) - ln (5)

= 3 ln (2) - ln (5)

What is the limit as t approaches 0 of (8^t-5^t)/t ?
The limit as t approaches 0 of (8^t-5^t)/t is 3ln(2)-ln(5)