Question:
True/False?
\( \lim_{n\to \infty } (n+1)^{1/3} -n^{1/3} = \infty \)
Hint:
Simplify the given expression.
Discussion:
We feel that \( (n+1)^{1/3} \) goes to infinity at the same speed as \( n^{1/3} \). So in fact, the above limit should be zero.
We make this little bit more rigorous.
\( (n+1)^{1/3} -n^{1/3} = \frac{n+1-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \)
\( =\frac{1}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \to 0\) as \(n\to \infty \).
