`lim_(u->oo) e^(u/10)/(u^3)` 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_(u->oo) e^(u/10)/(u^3)` 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 24 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the limit, hence, you need to replace `oo` for u:

`lim_(u->oo) (e^(u/10))/(u^3) = (e^oo)/(oo) = oo/oo`

Since the limit is indeterminate `oo/oo` , you may apply l'Hospital's rule:

`lim_(u->oo) (e^(u/10))/(u^3) = lim_(u->oo) ((e^(u/10))')/((u^3)')`

`lim_(u->oo) ((e^(u/10))')/((u^3)') = lim_(u->oo) ((1/10)*e^(u/10))/(3u^2) = oo/oo`

You need to use again l'Hospital's rule:

`lim_(u->oo) ((1/10)*e^(u/10))/((3u^2)) = lim_(u->oo) ((1/100)*e^(u/10))/(6u) = oo/oo`

You need to use again l'Hospital's rule:

`lim_(u->oo) ((1/1000)*e^(u/10))/6 = 1/6000*e^oo = oo`

Hence, evaluating the given limit using l'Hospital's rule, yields `lim_(u->oo) (e^(u/10))/(u^3) = oo.`

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