document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=182519>Question 182519</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me to find out <BR>\n" );
document.write( "lim (n,inf) 8n^2*pi*sqrt(n^2+n+1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #137043 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=137043\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n^2\"> in the radicand increases without bound as <i>n</i> increases without bound because:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to a} x^m = a^m\"> so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to \infty} x^m = \infty\">\r<BR>\n" );
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document.write( "Same thing for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\">  in the radicand because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n = n^1\">\r<BR>\n" );
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document.write( "The 1 in the radicand remains 1 regardless of the value of <i>n</i>, so we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to a} b = b\"> so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to \infty} b = b\">\r<BR>\n" );
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document.write( "But the limit of the sum is the sum of the limits, so if we say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math G(n) = n^2 + n + 1\">, then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{n\to \infty} G(n) = \infty + \infty + 1 = \infty\">\r<BR>\n" );
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document.write( "Now if we define <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large H(x) = sqrt(x) = x^{1 \over 2}\">, we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to \infty} H(x) = \infty\"> directly from\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to \infty} x^m = \infty\">\r<BR>\n" );
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document.write( "So that means\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{G(n)\to \infty} H(G(n)) = \infty\">\r<BR>\n" );
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document.write( "The factor of 8:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to \infty} b = b\"> so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{n\to \infty} 8 = 8\">\r<BR>\n" );
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document.write( "The factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \pi\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{x\to \infty} b = b\"> so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{n\to \infty} \pi = \pi\">\r<BR>\n" );
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document.write( "And finally the factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n^2\"> outside of the radicand:\r<BR>\n" );
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document.write( "As previously shown:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{n\to \infty} n^2 = \infty\">\r<BR>\n" );
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document.write( "The limit of the product is the product of the limits, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \lim_{n\to \infty} 8n^2\pi sqrt{n^2 + n + 1} = (\lim_{n\to \infty}8)(\lim_{n\to \infty}n^2)(\lim_{n\to \infty}\pi)(\lim_{n\to \infty}sqrt{n^2 + n + 1})\cr\cr\text{          } =(8)(\infty)(\pi)(\infty)=\infty\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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