<s>[INST] How do I evaluate this limit with an integral function? Could anybody give me some pointers on how do I evaluate this limit:
$$\lim_{t \to +\infty} \frac{\int_0^t x^9e^{-x^2}\,dx}{\int_0^t x^7e^{-x^2}\,dx}$$
I'm not asking for the complete solution, just a hint to point me in the right direction. I appreciate your help. Thanks.
 [/INST] $\frac{\int_0^tx^9e^{-x^2}dx}{\int_0^tx^7e^{-x^2}}=\frac{-\frac{1}{2}(e^{-t^2}(t^8+4t^6+12t^4+24t^2+24)+24)}{-\frac{1}{2}(e^{-t^2}(t^6+3t^4+6t^2+6)+6)}=\frac{\frac{1}{2}e^{-t^2}(t^8+4t^6+12t^4+24t^2+24+24e^{t^2})}{\frac{1}{2}(e^{-t^2}(t^6+3t^4+6t^2+6+6e^{t^2}))}=\frac{t^8+4t^6+12t^4+24t^2+24+24e^{t^2}}{t^6+3t^4+6t^2+6+6e^{t^2}}=\frac{e^{-t^2}(t^8+4t^6+12t^4+24t^2+24)+24}{e^{-t^2}(t^6+3t^4+6t^2+6)+6}\to 4\,\,\,,\,as\,t\to\infty$ 
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