[MATH]\lim_{n\to\infty}\frac{2n^2+4n^3}{n^3+5\sqrt{2+n^6}}[/MATH]
I could move a 2 out in front of the limit and then see if I could get one-third out of what's left but to no avail. L'Hospital's rule gives me a messy denominator and distributing the denominator's conjugate throughout I can already see that I'm going to get n's everywhere.
Which may be what this problem requires. For clarification's sake, this problem is not part of any assignment, rather I find limits interesting but in school only ever went as far as one semester of calculus.
To show I put brainpower behind this, here's what I get using L'Hospital:
[MATH]\lim_{n\to\infty}\frac{4n(1+3n)}{3n^2(1+\frac{5n^3}{\sqrt{2+n^6}})}[/MATH]
And messier after a second time, of course. And from dividing highest factor:
[MATH]\lim_{n\to\infty}\frac{2n^{-1}+4}{1+\sqrt{\frac{50+25n^6}{n^6}}}[/MATH]
I know the above could be cleaned up but I feel like I'm going in the wrong direction regardless and am hoping someone can provide me with a little nudge. Thanks in advance.
[MATH]\frac{2}{3}[/MATH]
I could move a 2 out in front of the limit and then see if I could get one-third out of what's left but to no avail. L'Hospital's rule gives me a messy denominator and distributing the denominator's conjugate throughout I can already see that I'm going to get n's everywhere.
Which may be what this problem requires. For clarification's sake, this problem is not part of any assignment, rather I find limits interesting but in school only ever went as far as one semester of calculus.
To show I put brainpower behind this, here's what I get using L'Hospital:
[MATH]\lim_{n\to\infty}\frac{4n(1+3n)}{3n^2(1+\frac{5n^3}{\sqrt{2+n^6}})}[/MATH]
And messier after a second time, of course. And from dividing highest factor:
[MATH]\lim_{n\to\infty}\frac{2n^{-1}+4}{1+\sqrt{\frac{50+25n^6}{n^6}}}[/MATH]
I know the above could be cleaned up but I feel like I'm going in the wrong direction regardless and am hoping someone can provide me with a little nudge. Thanks in advance.
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