\lim _(n\to \infty \:)\left(\sqrt[n]{(n^2+1)/(n+1)}\right) Please show your work! Thank you! :)

Question

`\lim _(n\to \infty \:)\left(\sqrt[n]{(n^2+1)/(n+1)}\right)`

Please show your work!
Thank you! :)

Answer 1

`\displaystyle \lim _(n\to \infty )\sqrt[n]{(n^2+1)/(n+1)} = \boxed1`

Explanation:

we want to compute the following limit:

`\displaystyle \lim _(n\to \infty )\sqrt[n]{(n^2+1)/(n+1)}`

well, remember that, for limits to infinity, terms less than the highest degree of the numerator or denominator can be disregarded, hence we can drop 1 which yields:

`\displaystyle \lim _(n\to \infty )\sqrt[n]{(n^2)/(n)}`

reduce fraction:

`\displaystyle \lim _(n\to \infty )\sqrt[n]{n}`

by using limit formula, we acquire:

`\displaystyle \lim _(n\to \infty )\sqrt[n]{n} = \boxed1`

and we're done!