lim（x→0）=(1/x^2-cot^2x)lim(x→∞)=[(2+x)e^(1/x)-x]

Question

lim（x→0）=(1/x^2-cot^2x)
lim(x→∞)=[(2+x)e^(1/x)-x]

Answer 1

lim（x→0）=(1/x^2-cot^2x)
=lim（x→0）=(1/x^2-1/tan²x)
=lim(x->0)(tan²x-x²)/x²tan²x
=lim(x->0)(tan²x-x²)/x^4
=lim(x->0)(tanx+x)(tanx-x)/x^4
=lim(x->0)(tanx+x)/x*lim(x->0)(tanx-x)/x³
=2lim(x->0)(sec²x-1)/3x²
=2lim(x->0)(tan²x)/3x²
=2/3
lim(x→∞)=[(2+x)e^(1/x)-x]
令x=1/t
原式=lim(t->0)[(2+1/t)e^t-1/t]
=lim(t->0)[(2t+1)e^t-1]/t
=lim(t->0)[(2t+1)e^t+2e^t]/1
=1+2
=3