Use this and only this fact to find lim (x->inf) (1+2/x+1/x^2)^x

use this and only this fact to find lim (x->inf) (1+2/x+1/x^2)^x
lim (x->inf) (1+1/x)^x = e

Note that 1 + 2/x + 1/x^2 = (1 + 1/x)^2, so you can write your limit as:
lim (x-->infinity) [(1 + 1/x)^2]^x = lim (x-->infinity) (1 + 1/x)^(2x).

Therefore, by the limit laws:
lim (x-->infinity) [(1 + 1/x)^2]^x = lim (x-->infinity) (1 + 1/x)^(2x)
= lim (x-->infinity) [(1 + 1/x)^x]^2
= [lim (x-->infinity) (1 + 1/x)^x]^2
= e^2, since lim (x-->infinity) (1 + 1/x)^x = e.

I hope this helps!

1 + 2/x + 1/x²
(1 + 1/x)(1 + 1/x)
(1 + 1/x)²

lim x→∞ ((1 + 1/x)²)^x
lim x→∞ ((1 + 1/x)^(2x))

Can be rewritten as
(lim x→∞ (1 + 1/x)^x) ∙ (lim x→∞ (1 + 1/x)^x)
e ∙ e
e²