I've got problems with calculating the limits in these two examples:
lim
Can anybody help?
Answer
For the first one, let u=\frac1{\sqrt n}, and note that u\to 0^+ as n\to\infty. Then \sqrt n\left(e^{\frac1{\sqrt n}}-2^{\frac1{\sqrt n}}\right)=\frac1u\left(e^u-2^u\right)=\frac{e^u-2^u}u\;, so you’re interested in \lim_{u\to 0^+}\frac{(e^u-2^u)^3}{u^3}\;, and I expect that you know a way to deal with that kind of limit.
I can make a similar trick work for the second one, but it gets a bit messier. First, I’m actually going to look at \lim_{n\to\infty}n^2\left(e^{\frac1n}-e^{\frac1{n+1}}\right) and then take its square root to get the desired limit.
Let u=\frac1n, so that n=\frac1u, n+1=\frac1u+1=\frac{u+1}u, and \frac1{n+1}=\frac{u}{u+1}=1-\frac1{u+1}. Again u\to 0^+ as n\to\infty, so I look at \lim_{u\to 0^+}\frac{e^u-e^{1-\frac1{u+1}}}{u^2}\;; applying l’Hospital’s rule takes a little more work this time, but it’s still eminently feasible.
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