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I can't seem to find this limit

1. The problem statement, all variables and given/known data
1.jpg

2. Relevant equations


3. The attempt at a solution
I tried using the rule of multiplying with the "conjugate", for example what's above multiplied by (√n^3+3n)+(√n^3+2n^2+3)/(√n^3+3n)+(√n^3+2n^2+3).
But I'm left with a huge mess :(
I also tried dividing the top and the bottom by n^2 in the square roots to get the n out, but that didn't work either :(
 

haruspex

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1. The problem statement, all variables and given/known data
View attachment 94117
2. Relevant equations


3. The attempt at a solution
I tried using the rule of multiplying with the "conjugate", for example what's above multiplied by (√n^3+3n)+(√n^3+2n^2+3)/(√n^3+3n)+(√n^3+2n^2+3).
But I'm left with a huge mess :(
I also tried dividing the top and the bottom by n^2 in the square roots to get the n out, but that didn't work either :(
The conjugate method looks good here. It doesn't give a mess in the numerator, right? In the denominator, do you need all the terms, or is it sufficient then only to look at the leading terms?
 
330
46
The image is so bad I can't even see the value of the exponents, try to write it out in latex next time.
Assuming what you wrote is ##\frac{\sqrt{n^3+3n}-\sqrt{n^3+2n^2+3}}{\sqrt{n+2}}##
multiplying with the conjugate works for me. After you done that note that only the "highest order" terms matter.
 

vela

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I also tried dividing the top and the bottom by n^2 in the square roots to get the n out, but that didn't work either :(
After you multiply by the conjugate, you want to pull the highest power of n, not ##n^2##, out of each of the square roots.
 

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