5.1 Computing Limits

An understanding of limits is fundamental to an understanding of calculus. Let s start by defining a few functions:

<span class="inlinemediaobject"><a NAME="IMG_453"> href="portalcontent.asp?bkid=31099&image_src=https://images.books24x7.com/bookimages/id_31099/fig207%5F01%5F0%2Ejpg&image_id=453&previd=IMG_453"> target="_parent"><img alt="Image from book"> border="0"> height="123"> id="IMG_453"> src="https://images.books24x7.com/bookimages/id_31099/fig207_01.jpg"> title="Click To expand"> width="156"></a></span>

Note that x = 0 is not in the domain of either of these functions. How do they behave as x approaches 0, that is, as x assumes values very close to 0? A plot is a sensible way to approach this question:

  In[4]:= <b class="bold">Plot[f[x], {x, <span class="unicode">?</span>10, 10}]</b><span class="inlinemediaobject"><a NAME="IMG_454"> href="portalcontent.asp?bkid=31099&image_src=https://images.books24x7.com/bookimages/id_31099/fig207%5F02%5F0%2Ejpg&image_id=454&previd=IMG_454"> target="_parent"><img alt="Image from book"> border="0"> height="148"> id="IMG_454"> src="https://images.books24x7.com/bookimages/id_31099/fig207_02.jpg"> title="Click To expand"> width="290"></a></span>  In[5]:= <b class="bold">Plot[g[x], {x, <span class="unicode">?</span>1, 1}]</b><span class="inlinemediaobject"><a NAME="IMG_455"> href="portalcontent.asp?bkid=31099&image_src=https://images.books24x7.com/bookimages/id_31099/fig207%5F03%5F0%2Ejpg&image_id=455&previd=IMG_455"> target="_parent"><img alt="Image from book"> border="0"> height="145"> id="IMG_455"> src="https://images.books24x7.com/bookimages/id_31099/fig207_03.jpg"> title="Click To expand"> width="290"></a></span>

The two outcomes are strikingly different, and they illustrate the likely possibilities for similar investigations. The function f( x) assumes values that approach 1 as x approaches 0. The function g has a vertical asymptote at x = 0; as x approaches 0 from the right, g assumes values that approach + ?, while as x approaches 0 from the left, g assumes values that approach ??.

We can check this numerically by making a table of values. Here is a table of values for f as x approaches 0 from the right:

  In[6]:= <b class="bold">data = Table[{N[10</b><sup<b class="bold"><span class="unicode">?</span>n</b></sup><b class="bold">], N[f[10</b><sup<b class="bold"><span class="unicode">?</span>n</b></sup><b class="bold">],...