Please help with this limit: [(3+n)(-sqrt[1+n]+sqrt[7+n])]/sqrt[4+n]

[emii64]

I need some help. I've tried so many different ways but ive never found the solution.

\(\displaystyle \displaystyle \lim_{n\, \rightarrow \, \infty}\, \dfrac{(3\, +\, n)\, \left(-\sqrt{\strut 1\, +\, n\,}\, +\, \sqrt{\strut 7\, +\, n\,}\right)}{\sqrt{\strut 4\, +\, n\,}}\)

 

[Deleted member 4993]

I need some help. I've tried so many different ways but ive never found the solution.

\(\displaystyle \displaystyle \lim_{n\, \rightarrow \, \infty}\, \dfrac{(3\, +\, n)\, \left(-\sqrt{\strut 1\, +\, n\,}\, +\, \sqrt{\strut 7\, +\, n\,}\right)}{\sqrt{\strut 4\, +\, n\,}}\)

Multiply by \(\displaystyle \displaystyle{\dfrac{\sqrt{1+n}+\sqrt{7+n}}{\sqrt{1+n}+\sqrt{7+n}}}\) - simplify - then take the limit

What are your thoughts?

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[emii64]

Thank you, i could find a solution

\(\displaystyle \dfrac{6n\, +\, 18}{\sqrt{\strut\, n^2\, +\, 5n\, +\, 4\,}\, +\, \sqrt{\strut \, n^2\, +\, 11n\, +\, 28\,}}\)

then

\(\displaystyle \dfrac{n\left(6\, +\, \dfrac{18}{n}\right)}{n\,\left(\, \sqrt{\strut\, \dfrac{11}{n}\, +\, \dfrac{28}{n^2}\, +\, 1\,}\, \right)\, +\, n\,\left(\, \sqrt{\strut\, 1\, +\, \dfrac{5}{n}\, +\, \dfrac{4}{n^2}\, }\, \right)}\, \)

\(\displaystyle =\, \dfrac{n\, \left(6\, +\, \dfrac{18}{n}\right)}{n\, \left( \sqrt{\strut\, \dfrac{11}{n}\, +\, \dfrac{28}{n^2}\, +\, 1\,}\, + \,1\, \sqrt{\strut\, 1\, +\, \dfrac{5}{n}\, +\, \dfrac{4}{n^2}\, }\, \right)}\)

and the limit is 3 . Its ok??

And sorry but im from Argentina and im not speak english very well.

 

[Deleted member 4993]

Thank you, i could find a solution

\(\displaystyle \dfrac{6n\, +\, 18}{\sqrt{\strut\, n^2\, +\, 5n\, +\, 4\,}\, +\, \sqrt{\strut \, n^2\, +\, 11n\, +\, 28\,}}\)

then

\(\displaystyle \dfrac{n\left(6\, +\, \dfrac{18}{n}\right)}{n\,\left(\, \sqrt{\strut\, \dfrac{11}{n}\, +\, \dfrac{28}{n^2}\, +\, 1\,}\, \right)\, +\, n\,\left(\, \sqrt{\strut\, 1\, +\, \dfrac{5}{n}\, +\, \dfrac{4}{n^2}\, }\, \right)}\, \)

\(\displaystyle =\, \dfrac{n\, \left(6\, +\, \dfrac{18}{n}\right)}{n\, \left( \sqrt{\strut\, \dfrac{11}{n}\, +\, \dfrac{28}{n^2}\, +\, 1\,}\, + \,1\, \sqrt{\strut\, 1\, +\, \dfrac{5}{n}\, +\, \dfrac{4}{n^2}\, }\, \right)}\)

and the limit is 3 . Its ok??

And sorry but im from Argentina and im not speak english very well.

Looks good to me!!