Evaluating $\;\lim_{n\to \infty} \left(1 + \frac{1}{(n^2 + n)}\right)^{ n^2 + n^{1/2}}$

Question

What is the value of $$\lim_{n\to \infty} \left(1 + \frac{1}{(n^2 + n)}\right)^{\large n^2 + n^{1/2}}\;\;?$$ By a basic assumption and induction i think it might be '$e$'. But how can this problem be evaluated?

Answer 1

Hint: If $a_n\to0$, $b_n\to\infty$ and $a_nb_n\to c$ then $(1+a_n)^{b_n}\to\mathrm e^c$. Your case is when $a_n=1/(n^2+n)$ and $b_n=n^2+\sqrt{n}$ hence $c=1$.

Can you show the general result?

Answer 2

Hints:

Note that $$\lim_{n \to \infty}(1+\frac1n)^n=e.$$

So $$\lim_{n \to \infty}(1+\frac{1}{n^2+n})^{(n^2+n)\frac{n^2+n^\frac12}{n^2+n}}=e^1=e.$$

Answer 3

Hint: start with the $\log$.

$$ \log u_n = (n^2 + \sqrt n)\log\left(1+\frac 1{n^2 + n}\right) $$and use $\log (1+u)\sim_{u\to 0} u$. To justify this manipulation, don't forget to mention the continuity of $\exp$ and $\log$.