Graphical+Representation+of+a+Limit

=GRAPHICAL REPRESENTATION OF A LIMIT= media type="youtube" key="ftB3FHW4EP4" height="385" width="480"

Before we start, I want to begin with a reminder of the delta-epsilon definition of limit (straight from wikipedia - [|http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit]). We will be dealing with this here by changing the delta-x.

Whenever a point //x// is within δ units of //P//, //f//(//x//) is within ε units of //L//

And now... we will delve into the topic!

Assuming a hole at x = .5, If you wanted to find the limit at x = .5

Start with a large delta-x, say .05 Calculate f(.5 - .05) and f(.5 + .05)

Then take a tenth of that delta-x: Calculate f(.5 - .005) and f(.5 + .005)

Then use a tenth of that delta-x, so... Calculate f(.5 - .0005) and f(.5 + .0005)

Continue this until you have a fairly certain idea of what the numbers are approaching from either side, in this case, the answer would be .25

Another situation...

Assuming that there is a hole at the peak (x = 3), Calculate the limit from both sides

Start with a large delta-x, say .30 Calculate f(3 - .30) and f(3 + .30)

then use a tenth of the same delta-x, so... Calculate f(3 - .030) and f(3 + .030)

Continue this until you have a good idea for what number the values are approaching from either side. In this case, both sides of the graph will approach the same value although you will not have the benefit of diffeerent approaches from the top and the bottom

Now take this curve for example:



For this graph we cannot use the same procedure to find the limit of f(x) as x approaches 0. For a start, we only have one side of the curve, so we cannot see what f(x) is approaching from both sides. To solve this, we look for the limit of f(x) as x approaches 0 from the right.

We would do something similar to this:

Calculate f(0 + .05) Calculate f(0 + .005) Calculate f(0 + .0005)

Pretty soon however, we see that this doesn't seem to be very productive. The numbers just keep getting higher! In this case, the limit is infinite. The f(x) values, as x approaches 0, are simply going up and up to infinity.

=**Now that we are on the topic of infinite limits... a word from the editor. When looking at a curve on the calculator, do not simply assume that it has an infinite limit if it seems to be going to infinity. Zoom out a little bit to make sure that the limit is not simply out of sight!!!**=

And now... an example of a limit that does not exist!

The limit as: x → x0+ ≠ x → x0-. Therefore, the limit as x → x0 does not exist.

ALGEBRAIC METHODS FOR EVALUATING A LIMIT AS A SOURCE OF COMPARISON
Example 1 Take the limit of the f(x) as x approaches 1


 * **f(0.9)** || **f(0.99)** || **f(0.999)** || **f(1)** || **f(1.001)** || **f(1.01)** || **f(1.1)** ||
 * **1.900** || **1.990** || **1.999** || **undefined** || **2.001** || **2.010** || **2.100** ||
 * As x approaches 1 from both sides f(x) gets closer and closer to 2, therefore making 2 the limit of f(x) as x approaches 1. The limit still exists even though f(1) does not exist because as x draws closer to 1 f(x) draws closer to 2 making it the limit to the function.

Example 2

What is the limit of f(x) as x approaches 2, f(x) = 2x + 6


 * **f(1.9)** || **f(1.99)** || **f(1.999)** || **f(2)** || **f(2.001)** || **f(2.01)** || **f(2.1)** ||
 * **9.800** || **9.980** || **9.998** || **10** || **10.002** || **10.020** || **10.200** ||
 * As x approaches 2 from both sides s(x) gets closer and closer to the value 10, therefore making 10 the limit of f(x) as x approaches 2.
 * As x approaches 2 from both sides s(x) gets closer and closer to the value 10, therefore making 10 the limit of f(x) as x approaches 2.

SUMMARY OF GRAPHICAL REPRESENTATION OF LIMITS VS. ALGEBRAIC REPRESENTATION
In general, a limit is a value that a function, f(x), stays close to as x stays close to another value. The general definition of a limit is that you can keep f(x) arbitrarily close to a certain value by keeping x close enough, but not equal to, another value. This is because the limit is a function value as you //approach// another value, and not really at another value. When dealing with graphs, it can be fairly simple to identify various limits, much simpler than resorting to any algebra. As you can see above, algebraically finding a limit can take a fair amount of time, whereas one look at a graph can provide instant results. If, as the graph approaches a designated x-value, the same f(x) value is approached from both sides, then the limit for that graph as x approaches that value is the f(x) value. Additionally, graphs can be utilized to find one sided limits, meaning the limits that graphs have as the approach certain x-values from the left or the right. Limits in graphs are great and can be a real confidence booster on tests because, odds are, you will get any question about them correct.