So what are limits? If you remember ONE thing from class today...I hope it's that limits are what functions are APPROACHING. NOT what they are equal to. Keep that in mind...especially as we start to explain the deeper meaning of limits during the next few classes.
I've included a few HW problems in the video below...(also use the HW hint sheet I gave you!)
Not that we would ever HAVE to do a limit problem like this...but the example below really illustrates the idea that we are interested in what number is being APPROACHED!
Numerical Approach to Limits
Example 1: Let f(x) = 2 x + 2 and compute f(x) as x takes values closer to 1. We first consider values of x approaching 1 from the left (x < 1).
| x | f(x) |
| 0.5 | 3 |
| 0.8 | 3.6 |
| 0.9 | 3.8 |
| 0.95 | 3.9 |
| 0.99 | 3.98 |
| 0.999 | 3.998 |
| 0.9999 | 3.9998 |
| 0.99999 | 3.99998 |
We now consider x approaching 1 from the right (x > 1).
| x | f(x) |
| 1.5 | 5 |
| 1.2 | 4.4 |
| 1.1 | 4.2 |
| 1.05 | 4.1 |
| 1.01 | 4.02 |
| 1.001 | 4.002 |
| 1.0001 | 4.0002 |
| 1.00001 | 4.00002 |
In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that limx→1 f(x) = 4.
NOTE: We are talking about the values that f(x) takes when x gets closer to 1 and not f(1). In fact we may talk about the limit of f(x) as x approaches a even when f(a) is undefined.
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