# Section 2 Limits (continued)

## 2.4 Limits on the Side

```           _
f(x) = Öx
```
In the real numbers we can only define it for  x ³ 0. This is illustrated in the graph. Notice that no function is graphed to the left of zero.

But what about the behavior of this function at zero? We do define  Ö0 = 0. But does this function have a limit at  x = 0? From our previous discussions, we would say that we have a limit at  x = 0  if, as x gets close to zero, Öx gets close to the limit, which, in this case, we would like to believe is zero as well. But isn't -0.0000001 close to zero? The problem is, -0.0000001 is negative, and Öx is undefined for  x < 0. With this function, x can only get close to zero from the positive side. And so we define a special one-sided limit, which we annotate as:

```           _
lim   Öx  =  0                                              eq. 2.4-1
x  > 0+
```
In contract form, this means that for any  e > 0  you might choose, no matter how small, I can find a  d > 0  small enough that
```     _
|Öx - 0|  £  e                                               eq. 2.4-2
```
whenever  0 < x - 0 £ d. Notice that we have removed the absolute value sign and resticted the difference to being positive in the "whenever" clause of the contract.

And in general, what we mean by (which is spoken as, "the limit of f(x) as x goes toward a from above.")

```    lim   f(x)  =  L                                            eq. 2.4-3
x  > a+
```
is the following contract: for any  e > 0  you might name, no matter how small, I can find a  d > 0 small enough that
```   |f(x) - L|  £  e                                             eq. 2.4-4
```
whenever  0 < x - a £ d. Note that this definition allows that x approach a only for values for which  x > a. This is in contrast to a standard limit in which x can approach a from either side and arrive at the same limit either way.

Of course, by symmetry, we might have another case in which we would like to approach the limit from below. So what we mean by (which is spoken as, "the limit of f(x) as x goes toward a from below.")

```    lim   f(x)  =  L                                            eq. 2.4-5
x  > a-
```
is the following contract: for any  e > 0  you might name, no matter how small, I can find a  d > 0  small enough that
```   |f(x) - L|  £  e                                             eq. 2.4-6
```
whenever  0 < a - x £ d. Notice that the only difference between this contract and the one in 2.4-4 is that we have turned the difference between x and a around in the "whenever" clause. And that resticts the contract to values of x that are less than a.

### Exercise

Prove that 2.4-1 is true using the contract in 2.4-2.

Move on to A Most Useful Rule

email me at hahn@netsrq.com