document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=85338>Question 85338</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Of Calculus I:<BR>\n" );
document.write( "Prove that the limit as x approaches zero from the right of the function<BR>\n" );
document.write( "\"The square root of x times e to the power of the sine of pi over x\"<BR>\n" );
document.write( "is equal to zero.<BR>\n" );
document.write( "<br><BR>\n" );
document.write( "In mathematical terms, that is <BR>\n" );
document.write( "Prove <BR>\n" );
document.write( "<tt><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( " lim <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29e%5E%28sin%28pi%2Fx%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29e%5E%28sin%28pi%2Fx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "x->0+\r\n" );
document.write( "</PRE></tt> </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #61582 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=61582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "lim <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29e%5E%28sin%28pi%2Fx%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29e%5E%28sin%28pi%2Fx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x->0+<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since nobody else has taken a shot at this, maybe I can give you a way of looking at it.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Look at the exponent of e and recognize the limits on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2Fx%29 ALIGN=MIDDLE  ALT=\"sin%28pi%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The sine function<BR>\n" );
document.write( "is limited to values between -1 and +1.  So the exponent of e will range in value from<BR>\n" );
document.write( "-1 to +1.  That means that the term:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28sin%28pi%2Fx%29%29 ALIGN=MIDDLE  ALT=\"e%5E%28sin%28pi%2Fx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "has a finite value ranging from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-1%29 ALIGN=MIDDLE  ALT=\"e%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28%2B1%29 ALIGN=MIDDLE  ALT=\"e%5E%28%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since this range of values<BR>\n" );
document.write( "is finite (and positive) the limiting factor in this problem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since<BR>\n" );
document.write( "x is approaching 0+, the expression gets closer and closer to zero times the finite value<BR>\n" );
document.write( "of the term involving e.  Therefore, the limit is zero, just as the problem proposes that<BR>\n" );
document.write( "you prove.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this gets you on the right track and at least gives you a feel for a way of proving the<BR>\n" );
document.write( "premise.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );