document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200146>Question 1200146</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate using limit laws: \r<BR>\n" );
document.write( "\n" );
document.write( "Limit (as x approaches negative infinity) (x/sqrt(x^2-1))\r<BR>\n" );
document.write( "\n" );
document.write( "Limit (as x approaches positive infinity) (x/sqrt(x^2-1)) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834181 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=834181\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "When x < 0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%29+=+-x ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%29+=+-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When x > 0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%29+=+x ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For very large positive or negative values of x, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This can be seen in the graph shown below<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/1llyy35ent\">https://www.desmos.com/calculator/1llyy35ent</a><BR>\n" );
document.write( "Desmos is a free graphing app.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The -1 at the end doesn't alter the radicand too much when x^2 is so very large.<BR>\n" );
document.write( "Examples:<BR>\n" );
document.write( "If x = 5, then x^2 = 25 and x^2-1 = 24<BR>\n" );
document.write( "If x = 50, then x^2 = 2500 and x^2-1 = 2499<BR>\n" );
document.write( "If x = 500, then x^2 = 250,000 and x^2-1 = 249,999<BR>\n" );
document.write( "If x = 5000, then x^2 = 25,000,000 and x^2-1 = 24,999,999\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for large negative values of x.<BR>\n" );
document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for large positive values of x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For very large negative values of x, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fsqrt%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"x%2Fsqrt%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28-x%29+=+-1 ALIGN=MIDDLE  ALT=\"x%2F%28-x%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For very large positive values of x, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fsqrt%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"x%2Fsqrt%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fx+=+1 ALIGN=MIDDLE  ALT=\"x%2Fx+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle \lim_{\text{x}\to -\infty} \left( \frac{\text{x}}{\sqrt{\text{x}^2-1}}\right) = -1\"><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle \lim_{\text{x}\to \infty} \left(\frac{\text{x}}{\sqrt{\text{x}^2-1}}\right) = 1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Verification using WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.wolframalpha.com/input?i=x%2Fsqrt%28x%5E2-1%29+asymptotes\">https://www.wolframalpha.com/input?i=x%2Fsqrt%28x%5E2-1%29+asymptotes</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Verification using Desmos<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/hfq1eravms\">https://www.desmos.com/calculator/hfq1eravms</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );