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Numerade Educator

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Problem 24 Easy Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{t\to 0} \frac{8^t - 5^t}{t} $

Answer

$\ln 8-\ln 5$

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Video Transcript

So for this problem, we are given the limit as T equals zero of 8 to the T -5 to the T over T. Um And if we plugged in zero we would obviously get 0/0. So as a result, we would want to apply lope towels rule. Now we know that the derivative of eight over T. Is going to end up giving us uh 8 to 8 to the teeth. The derivative will be eight to the T. Times the natural log of eight, And then this will be five to the T times the natural log of five. Then this right here is just going to become one when we take the derivative of that. So now when we plug in T equals zero, We get one times natural log of eight minus one times natural log of five. That's just going to end up giving us a final answer is natural log of eight minus natural log of five, or equally. The natural log of eight divided by five. It's about the same thing. Just based on log properties that we already know.