Numerically

Here is a table of [Graphics:../Images/index_gr_63.gif] values near [Graphics:../Images/index_gr_64.gif]: 

[Graphics:../Images/index_gr_65.gif]
[Graphics:../Images/index_gr_66.gif]
[Graphics:../Images/index_gr_67.gif]
0.9` 2.8976591363917925`
0.91` 2.9081140970280477`
0.92` 2.9185178511046184`
0.93` 2.928871232638975`
0.9400000000000001` 2.939175053104057`
0.9500000000000001` 2.9494301022730998`
0.9600000000000001` 2.959637149023939`
0.97` 2.969796942105475`
0.98` 2.9799102108684177`
0.99` 2.9899776659620425`
1.` Indeterminate
1.01` 3.0099778881942143`
1.02` 3.0199119889622215`
1.03` 3.029802944506765`
1.04` 3.039651381369445`
1.05` 3.049457910961346`
1.06` 3.0592231300700226`
1.07` 3.0689476213449187`
1.08` 3.0786319537626587`
1.09` 3.088276683072385`
1.1` 3.097882352222975`

This one might be more useful.  We use the [Graphics:../Images/index_gr_68.gif]to plug in decimals closer and closer to 1: 

[Graphics:../Images/index_gr_69.gif]
1.1` 3.0978823522229724`
1.01` 3.0099778881942116`
1.001` 3.0009997778875155`
1.0001` 3.000099997766138`
1.00001` 3.000010000014507`
1.000001` 3.0000010007801237`
1.0000001` 3.000000111491151`
1.00000001` 3.000000151459184`
1.000000001` 2.999999751778908`
1.0000000001` 2.9999997517789074`

Of course, we have to approach from the negative side as well: 

[Graphics:../Images/index_gr_70.gif]
0.9` 2.897659136391803`
0.99` 2.9899776659622384`
0.999` 2.998999777667013`
0.9999` 2.9998999977900764`
0.99999` 2.9999900000128523`
0.999999` 2.999999000380277`
0.9999999` 2.9999999216430115`
0.99999999` 3.0000000515391045`
0.999999999` 2.999999751778908`
0.9999999999` 3.000009743817756`
What does it look like the limit is here?

Converted by Mathematica      May 12, 2003