This limit would be quite easy to solve if you convert this into a fraction by multiplying and dividing by sqrt(n^2+n) + n.
Then, the limit would be converted to such limit:
limit as n approaches positive infinity, [n^2+n - n^2] / [sqrt(n^2+n) + n] = n / [sqrt(n^2+n) + n].
When you directly substitute positive infinity to n, you would get an indeterminate form of infinity over infinity. Therefore, we can use the L'Hôpital's rule to comput the limit. Hence, take the derivative of the numerator and denominator.
limit as n approaches positive infinity, 1 / [ (2n+1) / (2sqrt(n^2+n) ].
Now, you can use numerical analysis to comput the limit (especially the denominator part). As the leading power of the denominator and the numerator in the fraction of the overall limit's denominator is both 1, and the ratio of the leading coefficient is 2, we can numerically analyze that as n approaches positive infinity, the denominator of the whole expression approaches 2.
Therefore, the overall limit approaches 1/2 = 0.5.
Hi,
Thank you for your question.
This limit would be quite easy to solve if you convert this into a fraction by multiplying and dividing by sqrt(n^2+n) + n.
Then, the limit would be converted to such limit:
limit as n approaches positive infinity, [n^2+n - n^2] / [sqrt(n^2+n) + n] = n / [sqrt(n^2+n) + n].
When you directly substitute positive infinity to n, you would get an indeterminate form of infinity over infinity. Therefore, we can use the L'Hôpital's rule to comput the limit. Hence, take the derivative of the numerator and denominator.
limit as n approaches positive infinity, 1 / [ (2n+1) / (2sqrt(n^2+n) ].
Now, you can use numerical analysis to comput the limit (especially the denominator part). As the leading power of the denominator and the numerator in the fraction of the overall limit's denominator is both 1, and the ratio of the leading coefficient is 2, we can numerically analyze that as n approaches positive infinity, the denominator of the whole expression approaches 2.
Therefore, the overall limit approaches 1/2 = 0.5.
Does this answer your question?