Question

`lim_(x->+oo) (x)^2 (e^(1/(x))-e^(1/(x+1)))` ?

`lim_(x->+oo) color(white)(..)(x)^2 (e^(1/(x))-e^(1/(x+1)))` ?

Answer 1

`lim_(x->oo) x^2(e^(1/x)-e^(1/(x+1))) = 1`

Explanation:

Here's a slightly quick and dirty method using the series for `e^t`:

Note that:

`e^t = 1+t+t^2/2+t^3/6+...`

So:

`e^(1/x)-e^(1/(x+1)) = (1+1/x+1/(2x^2)+...)-(1+1/(x+1)+1/(2(x+1)^2)+...)`

`color(white)(e^(1/x)-e^(1/(x+1))) = 1/(x(x+1))+1/(O(x^3))`

So:

`lim_(x->oo) x^2(e^(1/x)-e^(1/(x+1))) = lim_(x->oo) (x^2/(x(x+1))+1/(O(x))) = 1`