`lim_(x->oo) (1 + a/x)^(bx)` 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,...

`lim_(x->oo) (1 + a/x)^(bx)` 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.

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Textbook Question

Chapter 4, 4.4 - Problem 58 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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tiburtius | High School Teacher | (Level 2) Educator

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We will use the fact that `lim_(x->oo)(1+1/x)^x=e.`

`lim_(x->oo)(1+a/x)^(bx)=lim_(x->oo)(1+a/x)^(bx/a cdot a)`

Now we use substitution `y=x/a.`

`=lim_(y->oo)(1+1/y)^(aby)`

`=lim_(y->oo)((1+1/y)^y)^(ab)`

`=(lim_(y->oo)(1+1/y)^y)^(ab)`

`=e^(ab)`

Therefore, the solutions is `e^(ab).`

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