`lim_(x->oo) x^(ln(2)/(1 + ln(x)))` 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...

`lim_(x->oo) x^(ln(2)/(1 + ln(x)))` 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 60 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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Given the limit function `lim_{x->infty}x^((ln(2)/(1+ln(x))` . We have to find the limit value.

This is of the form `infty^0`  and can be written as:

`lim_{x->infty}x^((ln(2))/(1+ln(x)))=e^(lim_{x->infty}ln(2)/(1+ln(x))ln(x))`

So,

`lim_{x->infty}ln(2)/(1+ln(x))ln(x)=lim_{x->infty}ln(2)/[ln(x)(1/ln(x)+1)]ln(x)`

                            `=lim_{x->infty}ln(2)/(1/ln(x)+1)`

                            `=ln(2)`

Therefore the limit value is `e^(ln(2))=2`

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