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lim(t to ∞)[(t^2)/e^t]
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\[\lim_{t\to\infty}(t^2)/e^t = \infty/\infty\] which permits you to use l'hopital's rule \[\lim_{t\to\infty}(2t)/e^t = \infty/\infty\] which permits u to use l'hopital's rule, again so; \[\lim_{t\to\infty}(2)/e^t = 0\] finally an answer.
t^2/e^t t^2 =-------- 1+t+t^2/2!+..... 1 =-------- t^-2 +t^-1+1/2!+t/3!...... 1 =----------- 0+0+0.5+t/6+..... =0 when t->infty
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