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### Limits of more complicated functions

It should be no secret by now that for most functions defined by a single formula, when exists. For more complicated functions, this may not be true. For dealing with some of these exceptional cases, we need to define right-hand and left-hand limits. Loosely speaking, the right-hand limit of at a is L if approaches L as x approaches a from the right. That is, the values of x satisfy x > a. The left-hand limit is defined in an analogous manner, with the values of x approaching a from the left. Maple can compute these special limits with commands like those shown below. The Maple floor function is actually the greatest integer function.

```  > plot(floor(x),x=0..4);
```

```  > limit(floor(x),x=1,right);
```

```  > limit(floor(x),x=1,left);
```

The floor function is one of Maple's defined functions, so you might expect things to work properly. If you define your own function in a piecewise fashion, however, a different approach is needed. For example, suppose you wanted to find the limit as x approaches 2 for the following function:

You would need to execute the following commands:

```  > f1 := x -> sqrt(2-x);
```

```  > limit(f1(x),x=2,left);
```

```  > f2 := x -> x;
```

```  > limit(f2(x),x=2,right);
```

From this, we see that the limit of f as x approaches 2 does not exist since the limit from the left does not equal the limit from the right.

William W. Farr
Thu Sep 7 15:27:05 EDT 1995