If 0≤t≤x, then 1+t4≤1+x4 , hence
1=1+041≥1+t41≥1+x41, hence
31=x31∫0xt2dt≥x31∫0x1+t4t2dt≥x31∫0x1+x4t2dt=3(1+x4)1,
x→+0lim31≥x→+0limx31∫0x1+t41dt≥x→+0lim3(1+x4)1=3(1+0)1=31,
By The Squeeze Theorem
x→+0limx31∫0x1+t4t2dt=31.
Similarly, if x≤t≤0, then 1+t4≤1+x4 ,
hence
1=1+041≥1+t41≥1+x41,
hence for x<0
31=x31∫0xt2dt=−x31∫x0t2dt≥−x31∫x01+t4t2dt=x31∫0x1+t4t2dt≥−x31∫x01+x4t2dt=x31∫0x1+x4t2dt=3(1+x4)1,
hence
x→−0lim31≥x→−0limx31∫0x1+t41dt≥x→−0lim3(1+x4)1=3(1+0)1=31,
by The Squeeze Theorem
x→−0limx31∫0x1+t4t2dt=31,
hence
x→0limx31∫0x1+t41dt=31.
Answer: 31.
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