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Numerade Educator

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Problem 60 Hard 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_{x\to \infty} \left( 1 + \frac{a}{x} \right)^{bx} $

Answer

$y=e^{a b}$

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Video Transcript

here we want to take the limit as X goes to infinity of the function. The function we're given is one plus A over X to the power of the X. So then we can take the natural log of both sides and simplify further. Want to get us um we'll end up having be times natural log of one plus A over X over one over X. Which allows us to let T equal one over X. So we'll replace this here will make that 80 and now T is going to be approaching zero instead of X approaching infinity. That's going to give us an indeterminate form. So we'll use the hotel's rule giving us a B Over one plus 80. Now when we plug in T approaching zero, we just get the natural log of Y equals A B. So why is going to equal E to the A. B. Final answer?