求极限如图
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let y=1/x
lim(x->+∞) ∫ (1->x) [ t^2.(e^(1/t) -1) -t ] dt / [x^2. ln(1+ 1/x)]
=lim(x->+∞) ∫ (1->x) [ t^2.(e^(1/t) -1) -t ] dt / [x^2. (1/x)]
=lim(x->+∞) ∫ (1->x) [ t^2.(e^(1/t) -1) -t ] dt / x (∞/∞)
=lim(x->+∞) [ x^2.(e^(1/x) -1) -x ] / x
=lim(x->+∞) [ x(e^(1/x) -1) -1 ]
=lim(y->0) [ (e^y -1) -y ] /y (0/0)
=lim(y->0) [ e^y -1 ]
=0
lim(x->+∞) ∫ (1->x) [ t^2.(e^(1/t) -1) -t ] dt / [x^2. ln(1+ 1/x)]
=lim(x->+∞) ∫ (1->x) [ t^2.(e^(1/t) -1) -t ] dt / [x^2. (1/x)]
=lim(x->+∞) ∫ (1->x) [ t^2.(e^(1/t) -1) -t ] dt / x (∞/∞)
=lim(x->+∞) [ x^2.(e^(1/x) -1) -x ] / x
=lim(x->+∞) [ x(e^(1/x) -1) -1 ]
=lim(y->0) [ (e^y -1) -y ] /y (0/0)
=lim(y->0) [ e^y -1 ]
=0
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