Límites unilaterales

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B>C á l c u l o   2 1</B><FONT SIZE="4" COLOR="#FFCC66" FACE="Bell MT" ><B> </B></FONT><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" ><B>   </B></FONT><B>  |   </B><A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U>Home</U></A>  |  <A HREF="id98.htm" TARGET="_top" TITLE="Contenido"><U>Contenido</U></A>  |  <A HREF="id293.htm" TARGET="_top" TITLE="Límites y continuidad"><U>Límites y continuidad</U></A>  |</div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> <bR> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Límites unilaterales</div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3926688482" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B> <A NAME="límites_unilaterales"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>H</B></FONT>ay casos en que las funciones no están definidas (en los reales) a la izquierda o a la derecha de un número determinado, por lo que el límite de la función cuando <I>x</I> tiende a dicho número, que supone que existe un intervalo abierto que contiene al número, no tiene sentido. </div> <div> <B>Ejemplo: </B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" > </FONT></div> <div> <IMG SRC="10e3537b0.gif" border=0 width="595" height="123" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="l_mite_unilateral_por_la_derecha"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Límite unilateral por la derecha:</B></FONT></div> <div> Sea <I>f</I> una función definida en todos los números del intervalo abierto (<I>a, c</I>). Entonces, el límite de <I>f </I>(<I>x</I>), cuando <I>x</I> se aproxima a <I>a</I> por la derecha es <I>L</I>, y se escribe</div> <div> <IMG SRC="10e405510.gif" border=0 width="517" height="81" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="l_mite_unilateral_por_la_izquierda"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Límite unilateral por la izquierda:</B></FONT></div> <div> Sea <I>f</I> una función definida en todos los números de (<I>d, a</I>). Entonces, el límite de <I>f </I>(<I>x</I>), cuando <I>x</I> se aproxima a <I>a</I> por la izquierda es <I>L</I>, y se escribe</div> <div> <IMG SRC="10e505510.gif" border=0 width="517" height="81" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="limite_bilateral"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="10e651300.gif" border=0 width="593" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  </div> <div>  <A NAME="teorema_de_l_mite9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema de límite12:</B></FONT></div> <div> <B><IMG SRC="10e771210.gif" border=0 width="625" height="33" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="612" BORDERCOLORLIGHT="#006600" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#FFFFCC HEIGHT= 58 WIDTH="602"><div>  <A NAME="ejercicios_resueltos"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>Ejercicios resueltos</B></FONT></div> <div> <B>          E</B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >n los ejercicios 1 a 4, trace la gráfica y determine el límite indicado si existe; si no existe, dé la razón:</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 100 ><NOBR><div> <A HREF="#1__" TITLE="Límites unilaterales"><U><IMG SRC="10e8215d0.gif" border=0 width="289" height="93" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#2__" TITLE="Límites unilaterales"><U><IMG SRC="10e92d4b0.gif" border=0 width="301" height="75" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 84 ><NOBR><div> <A HREF="#3__" TITLE="Límites unilaterales"><U><IMG SRC="10ea254d0.gif" border=0 width="293" height="77" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="301"><div> <A HREF="#4__" TITLE="Límites unilaterales"><U><IMG SRC="10ebb93b0.gif" border=0 width="185" height="59" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> </TABLE></div> <div> <B>S o l u c i o n e s</B></div> <div>  <A NAME="1__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>1. </B></FONT>Solución:</div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="610" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 207 ><NOBR><div> <IMG SRC="10ec64670.gif" border=0 width="356" height="103" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="10edc8c80.gif" border=0 width="200" height="200" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="2__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>2. </B></FONT>Solución:</div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="628" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 232 ><NOBR><div> <IMG SRC="10ee17850.gif" border=0 width="279" height="133" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="10efe1e10.gif" border=0 width="225" height="225" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="3__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>3. </B></FONT>Solución:</div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="619" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 207 ><NOBR><div> <IMG SRC="10f018640.gif" border=0 width="280" height="100" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="10f12cc80.gif" border=0 width="300" height="200" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="4__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>4. </B></FONT>Solución:</div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="617" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 242 ><NOBR><div> <IMG SRC="10f213630.gif" border=0 width="275" height="99" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="10f32ceb0.gif" border=0 width="300" height="235" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <bR> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B> <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:4577786115" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="100" /> <input type="submit" name="sa" value="Buscar" /> </div> </form> <script type="text/javascript" src="http://www.google.com.co/coop/cse/brand?form=cse-search-box&lang=es"></script>  </B></div> </td> </tr></table></body>