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Evaluate what is limit of squareroot(n+1)*(5^(1/squareroot n) -1), n goes to infinitte

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andraoida | Student, Grade 10 | (Level 2) eNoter

Posted November 25, 2011 at 2:22 AM via web

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Evaluate what is limit of squareroot(n+1)*(5^(1/squareroot n) -1), n goes to infinitte

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 25, 2011 at 2:44 AM (Answer #1)

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Use the formula `lim_(n->oo)` `(a^n-1)/n=ln a`

`lim_(n->oo)` `(sqrt(n+1))*((5^(1/sqrt n) - 1)/(1/sqrt n))*(1/sqrt n)` = `lim_(n->oo)``(5^(1/sqrt n)-1)/(1/sqrt n)* ` `lim_(n->oo)` `sqrt(n+1)/(sqrt n)`

`lim_(n->oo)``(sqrt n)*sqrt(1 + (1/n))/ sqrt n`  = `lim_(n-gtoo) sqrt (1 + (1/n))` = 1

`lim_(n-gtoo) (sqrt(n+1))*(5^(1/sqrt n) - 1) = ln 5`

ANSWER: After evaluation, the value of the limit of the function is ln 5.

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