What is the limit as t approaches 0 of (4^t-3^t)/t ?

The limit as t approaches 0 of (4^t-3^t)/t is 2ln(2)-ln(3)

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limit as t approaches 0 of (4^t-3^t)/t

t \to 0 lim (\frac{4 ^{t} - 3 ^{t}}{t})

2 ln (2) - ln (3)

Decimal Notation

0.28768 \dots

Solution steps

t \to 0 lim (\frac{4 ^{t} - 3 ^{t}}{t})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{ln ( 2 ) \cdot 2 ^{2 t + 1} - 3 ^{t} ln ( 3 )}{1})

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

Plug in the value

t = 0

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

Simplify

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

= 2 ln (2) - ln (3)

x \to 0 - lim (\frac{x ^{3} - 2}{x})

x \to \infty lim (\frac{5 x ^{4} + 2 x}{x ^{2}})

x \to 0 lim (\frac{( 1 - x ) ^{3} - 1}{x})

x \to 3 lim (2 + 3 x^{2})

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

What is the limit as t approaches 0 of (4^t-3^t)/t ?
The limit as t approaches 0 of (4^t-3^t)/t is 2ln(2)-ln(3)