`lim_(x->0) (x3^x)/(3^x - 1)` 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->0) (x3^x)/(3^x - 1)` 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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Chapter 4, 4.4 - Problem 31 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`lim_(x->0)(x3^x)/(3^x-1)`

Apply L'Hospital rule , Test condition:0/0

`=lim_(x->0)((x3^x)')/((3^x-1)')`

`=lim_(x->0)(x3^xln(3)+3^x)/(3^xln(3))`

`=lim_(x->0)(3^x(xln(3)+1))/(3^xln(3))`

`=lim_(x->0)(xln(3)+1)/ln(3)`

plug in the value,

`=(0ln(3)+1)/ln(3)`

`=1/ln(3)`

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