Answer to Question #131236 in Calculus for Moel Tariburu

Question #131236
Evaluate the limits:
lim┬(x→0)⁡〖1/x^3 〗 ∫_0^x▒〖t^2/(t^4+1) dt〗
1
Expert's answer
2020-08-31T16:39:35-0400

If 0tx0\leq t\leq x, then 1+t41+x41+t^4\leq 1+x^4 , hence

1=11+0411+t411+x4,1=\frac{1}{1+0^4}\geq\frac{1}{1+t^4}\geq\frac{1}{1+x^4}, hence

13=1x30xt2dt1x30xt21+t4dt1x30xt21+x4dt=13(1+x4)\frac{1}{3}=\frac{1}{x^3}\intop_0^x t^2dt\geq\\ \frac{1}{x^3}\intop_0^x\frac{t^2}{1+t^4}dt\geq\frac{1}{x^3}\intop_0^x\frac{t^2}{1+x^4}dt=\frac{1}{3(1+x^4)},

limx+013limx+01x30x11+t4dtlimx+013(1+x4)=13(1+0)=13,\lim\limits_{x\rightarrow+0}\frac{1}{3}\geq\\ \lim\limits_{x\rightarrow+0}\frac{1}{x^3}\intop_0^x\frac{1}{1+t^4}dt\geq\\ \lim\limits_{x\rightarrow+0}\frac{1}{3(1+x^4)}=\frac{1}{3(1+0)}=\frac{1}{3},

By The Squeeze Theorem

limx+01x30xt21+t4dt=13.\lim\limits_{x\rightarrow+0}\frac{1}{x^3}\intop_0^x\frac{t^2}{1+t^4}dt=\frac{1}{3}.

Similarly, if xt0x\leq t\leq 0, then 1+t41+x41+t^4\leq 1+x^4 ,

hence

1=11+0411+t411+x4,1=\frac{1}{1+0^4}\geq\frac{1}{1+t^4}\geq\frac{1}{1+x^4},

hence for x<0x<0

13=1x30xt2dt=1x3x0t2dt1x3x0t21+t4dt=1x30xt21+t4dt1x3x0t21+x4dt=1x30xt21+x4dt=13(1+x4),\frac{1}{3}=\frac{1}{x^3}\intop_0^x t^2dt=-\frac{1}{x^3}\intop_x^0 t^2dt\geq\\ -\frac{1}{x^3}\intop_x^0\frac{t^2}{1+t^4}dt=\frac{1}{x^3}\intop_0^x\frac{t^2}{1+t^4}dt\geq-\frac{1}{x^3}\intop_x^0\frac{t^2}{1+x^4}dt=\\ \frac{1}{x^3}\intop_0^x\frac{t^2}{1+x^4}dt=\frac{1}{3(1+x^4)},

hence

limx013limx01x30x11+t4dtlimx013(1+x4)=13(1+0)=13,\lim\limits_{x\rightarrow-0}\frac{1}{3}\geq\\ \lim\limits_{x\rightarrow-0}\frac{1}{x^3}\intop_0^x\frac{1}{1+t^4}dt\geq\\ \lim\limits_{x\rightarrow-0}\frac{1}{3(1+x^4)}=\frac{1}{3(1+0)}=\frac{1}{3},

by The Squeeze Theorem

limx01x30xt21+t4dt=13,\lim\limits_{x\rightarrow-0}\frac{1}{x^3}\intop_0^x\frac{t^2}{1+t^4}dt=\frac{1}{3},

hence

limx01x30x11+t4dt=13.\lim\limits_{x\rightarrow0}\frac{1}{x^3}\intop_0^x\frac{1}{1+t^4}dt=\frac{1}{3}.

Answer: 13.\frac{1}{3}.


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