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

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

##### 1 Answer

#### Explanation:

Here's a slightly quick and dirty method using the series for

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#