Evaluate a limit

LoudBar

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Please help. The answer is in the brackets, but how to get there...

1600192953132.png
 

firemath

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What have you tried? Where are you stuck? If you tell us these things, we can help you better.
 

firemath

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S/he wants the process.....?
 

firemath

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Moving from non-bracketed problem 15 to the \(\displaystyle 1/3\) in brackets? Maybe?
 

skeeter

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I assume you're trying to determine \(\displaystyle \lim_{n \to \infty} \{ x_n \}\)

note ...

\(\displaystyle (a-b)(a^2+ab+b^2) = a^3-b^3\)

let \(\displaystyle a = \sqrt[3]{1+\dfrac{2}{n}}\) and \(\displaystyle b = 1\)

... multiply your expression by \(\displaystyle \dfrac{a^2+ab+b^2}{a^2+ab+b^2}\)
 

LoudBar

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I assume you're trying to determine \(\displaystyle \lim_{n \to \infty} \{ x_n \}\)

note ...

\(\displaystyle (a-b)(a^2+ab+b^2) = a^3-b^3\)

let \(\displaystyle a = \sqrt[3]{1+\dfrac{2}{n}}\) and \(\displaystyle b = 1\)

... multiply your expression by \(\displaystyle \dfrac{a^2+ab+b^2}{a^2+ab+b^2}\)
Yeah, i tried it and idk where is my mistake or what i suppose to do next
 

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Subhotosh Khan

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Yeah, i tried it and idk where is my mistake or what i suppose to do next
Your work is correct.

Now take the limit - the denominator should go to 3 as n \(\displaystyle \to \infty \) ..... [& 1/n \(\displaystyle \to \) 0]
 

LoudBar

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Your work is correct.

Now take the limit - the denominator should go to 3 as n \(\displaystyle \to \infty \) ..... [& 1/n \(\displaystyle \to \) 0]
I think i got it, thank You a lot for helping me!
 

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Jomo

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lookagain

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Please help. The answer is in the brackets, but how to get there...

View attachment 21654
Here are a couple of approximations that may help you sometimes:

\(\displaystyle \sqrt{a + b} \ \approx \ \sqrt{a} + \dfrac{1}{2}b\)

\(\displaystyle \sqrt[3]{a + b} \ \approx \ \sqrt[3]{a} + \dfrac{1}{3}b\)

For this problem, a = 1 and b = 2/n for the cube root approximation.

The expression then is approximated by

\(\displaystyle \dfrac{n}{2}\bigg(1 + \dfrac{2}{3n} \ - \ 1 \bigg)\)

Simplify that and then take the limit as n approaches infinity.
 
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