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Limits: A Graphical Approach - Problem 2
Using graphs to find limits of functions is a visual exercise. To find the limit of f(x) as x approaches a certain point, observe what value f(x) gets close to as x approaches that point from the left and from the right. The limit exists if the limits from the left and the right are the same (when x is less than the point and when x is greater than the point, respectively). It is important to realize that the limit of a function at a certain point is a different matter from the value of the function at that point.
Let’s take a look at another problem. Same function f(x). I want to answer these questions: what is the limit as x approaches 3 from the left, as x approaches 3 from the right, as x approaches 3? And what’s the value of the function f(3)?
Here’s x equals 3. As we approach 3 from the left, we’re on this upward slanted part of the line. We’re going up upwards as x approaches 3 towards this point. And the y coordinate of that point is 4. So f(x) is approaching 4.
Now as x approaches 3 from the right side, we’re on this branch here. So as x approaches 3, we’re going up this curve and the y values are also approaching 4.
Now you’ll recall that the two sided limit of f(x), as x approaches 3, only exists if the two one sided limits exist and are equal. If they are equal then this becomes the value of the two sided limit. 4.
What about f(3)? Well notice that the limit as x approaches 3 of f(x) is 4. But f(3) is not 4. This point here represents the value of f(3). When x equals 3, f(3) is 1.
Keep it separate in your mind what actually happens to a function at 3, versus what happens as x approaches 3. They are completely different things. Later on though, we’ll see that there’s a connection for certain kinds of function called continuous functions.