Límites infinitos

<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 infinitos</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>E</B></FONT>xisten ciertas funciones que aumentan o disminuyen sin límite a medida que la variable independiente se acerca a un valor fijo determinado.</div> <bR> <div>  <A NAME="crecimiento_infinito"><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>Crecimiento infinito:</B></FONT> </div> <div> <IMG SRC="10f47d6b0.gif" border=0 width="637" height="107" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="decrecimiento_infinito"><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>Decrecimiento infinito:</B></FONT></div> <div> <IMG SRC="10f57d6b0.gif" border=0 width="637" height="107" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_de_límite13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema de límite13:</B></div> <div> <IMG SRC="10f617640.gif" border=0 width="279" height="100" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_de_límite14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema de límite14:</B></div> <div> <IMG SRC="10f7fd0f0.gif" border=0 width="509" height="271" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_de_límite15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema de límite15:</B></div> <div> <IMG SRC="10f8e36b0.gif" border=0 width="483" height="107" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_de_límite16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema de límite16:</B></div> <div> <IMG SRC="10f9b75b0.gif" border=0 width="439" height="91" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_de_lìmite_17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema de lìmite 17:</B></div> <div> <B><IMG SRC="10fab75b0.gif" border=0 width="439" height="91" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>      <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>U</B></FONT>na <FONT SIZE="3" COLOR="#FF6633" FACE="Times New Roman" ><I>asíntota</I></FONT> es una recta a la cual se aproxima una curva indefinidamente. Trazar las asíntotas, tanto verticales como horizontales (más adelante nos ocuparemos de estas últimas), es de gran ayuda para dibujar la gráfica de una función. </div> <bR> <div>  <A NAME="as_ntota_vertical"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#FF9999" FACE="Times New Roman" ><B>Asíntota vertical:</B></FONT></div> <div> Una asíntota vertical es una recta paralela al eje <I>y</I>.</div> <div> Se dice que la recta <I>x </I>=<I> a</I> es una <I>asíntota vertical</I> de la gráfica de la función  <I>f</I>  si por lo menos uno de los siguientes enunciados es verdadero:</div> <bR> <div> <IMG SRC="10fb898b0.gif" border=0 width="137" height="139" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="615" BORDERCOLORLIGHT="#33FFCC" BORDERCOLORDARK="#339900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFFFCC HEIGHT= 76 WIDTH="603"><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="#CC0033" FACE="Times New Roman" ><B>Ejercicios resueltos</B></FONT></div> <bR> <div> <FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><B>E</B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >n los ejercicios </FONT>1<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" > a </FONT>7<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >, determine el límite. En los ejercicios </FONT>9<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" > a </FONT>11<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >. encuentre la(s) asíntota(s) vertical(es) de la gráfica de la función y trácela(s).</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="199"><div> <A HREF="#1__" TITLE="Límites infinitos"><U><IMG SRC="10fc5c2b0.gif" border=0 width="92" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="199"><div> <A HREF="#2__" TITLE="Límites infinitos"><U><IMG SRC="10fd6d300.gif" border=0 width="109" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="199"><div> <A HREF="#3__" TITLE="Límites infinitos"><U><IMG SRC="10fe742d0.gif" border=0 width="116" height="45" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 54 ><NOBR><div> <A HREF="#4__" TITLE="Límites infinitos"><U><IMG SRC="10ffbb2d0.gif" border=0 width="187" height="45" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="199"><div> <A HREF="#5_" TITLE="Límites infinitos"><U><IMG SRC="1100702b0.gif" border=0 width="112" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#6__" TITLE="Límites infinitos"><U><IMG SRC="1101892f0.gif" border=0 width="137" height="47" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 96 ><NOBR><div> <IMG SRC="1102982d0.gif" border=0 width="152" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=2><NOBR><div> <IMG SRC="11038f590.gif" border=0 width="399" height="89" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 52 WIDTH="199"><div> <IMG SRC="110467290.gif" border=0 width="103" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="199"><div> <IMG SRC="11057f2d0.gif" border=0 width="127" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="110694290.gif" border=0 width="148" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></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="#CC0033" FACE="Times New Roman" ><B>1</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>. </B></FONT><U>Solución</U>:</div> <div> <IMG SRC="110761950.gif" border=0 width="609" height="149" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="#CC0033" FACE="Times New Roman" ><B>2. </B></FONT><U>Solución</U>:</div> <div> <IMG SRC="110837bb0.gif" border=0 width="311" height="187" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="#CC0033" FACE="Times New Roman" ><B>3.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT><U>Solución</U>:</div> <div> <IMG SRC="110992b30.gif" border=0 width="402" height="179" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="#CC0033" FACE="Times New Roman" ><B>4.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT><U>Solución</U>:</div> <div> <IMG SRC="110a2bf80.gif" border=0 width="555" height="504" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="5_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>5.</B></FONT> <U>Solución</U>:</div> <div> <IMG SRC="110bfdc70.gif" border=0 width="509" height="199" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="6__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>6.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT><U>Solución</U>:</div> <div> <IMG SRC="110c5e1d0.gif" border=0 width="606" height="285" ALT="Documento Microsoft Office Word" 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