lim t approaching infinity [t ln(1+3/t)

Answer 1

Since t →∞, (3/t) → 0

So using Taylor expansion, ln(1+3/t) = 3/t - (3/t)²/2 + (3/t)³/3 - ...

The expression t*ln(1+3/t) = 3 - O(1/t)

So, lim t →∞ t*ln(1+3/t) = 3