What is #lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3# ?

1 Answer
George C.
Dec 1, 2017

#lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3 = 3#

Explanation:

#lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3 = lim_(t->oo) t ln((t+1)/t)^3#

#color(white)(lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3) = lim_(t->oo) t * 3 ln(1+1/t)#

#color(white)(lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3) = lim_(s->0^+) 3 * 1/s ln(1+s)#

#color(white)(lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3) = lim_(s->0^+) 3 * 1/s (s-s^2/2+s^3/3-s^4/4+...)#

#color(white)(lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3) = lim_(s->0^+) 3 * (1-s/2+s^2/3-s^3/4+...)#

#color(white)(lim_(x->-oo) x^2 ln((x^2+1)/x^2)^3) = 3#

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