Limits (Formal Definition) Please read Introduction to Limits first Approaching ... Sometimes we cant work something out directly ... but we can see what it should be as we get closer and closer! Example: (x² − 1)/(x − 1) Lets work it out for x=1: (12 − 1)/(1 − 1) = (1 − 1)/(1 − 1) = 00 Now 0/0 is a difficulty! We dont really know the value of 0/0 (it is indeterminate), so we need another way of answering this. So instead of trying to work it out for x=1 lets try approaching it closer and closer: Example Continued: x (x2 − 1)/(x − 1) 0.5 1.50000 0.9 1.90000 0.99 1.99000 0.999 1.99900 0.9999 1.99990 0.99999 1.99999 ... ... Now we see that as x gets close to 1, then (x²−1)/(x−1) gets close to 2 We are now faced with an interesting situation: When x=1 we dont know the answer (it is indeterminate) But we can see that it is going to be 2 We want to give the answer 2 but cant, so instead mathematicians say exactly what is going on by using the special word limit The limit of (x²−1)/(x−1) as x approaches 1 is 2 And it is written in symbols as: So it is a special way of saying, ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2 As a graph it looks like this: So, in truth, we cannot say what the value at x=1 is. But we can say that as we approach 1, the limit is 2. More Formal But instead of saying a limit equals some value because it looked like it was going to, we can have a more formal definition. So lets start with the general idea. From English to Mathematics Lets say it in English first: f(x) gets close to some limit as x gets close to some value When we call the Limit L, and the value that x gets close to a we can say f(x) gets close to L as x gets close to a Calculating Close Now, what is a mathematical way of saying close ... could we subtract one value from the other? Example 1: 4.01 − 4 = 0.01 Example 2: 3.8 − 4 = −0.2 Hmmm ... negatively close? That doesnt work ... we really need to say I dont care about positive or negative, I just want to know how far which is the absolute value. How Close = |a−b| Example 1: |4.01−4| = 0.01 Example 2: |3.8−4| = 0.2 And when |a−b| is small we know we are close, so we write: |f(x)−L| is small when |x−a| is small Needs Flash Player And this animation shows what happens with the function f(x) = (x2−1)(x−1) as x approaches a=1, f(x) approaches L=2 So |f(x)−2| is small when |x−1| is small. Delta and Epsilon But small is still English and not Mathematical-ish. Lets choose two values to be smaller than: that |x−a| must be smaller than that |f(x)−L| must be smaller than (Note: Those two greek letters, δ is delta and ε is epsilon, are often used for this, leading to the phrase delta-epsilon) And we have: |f(x)−L|0, there is a >0 so that |f(x)−L|
Posted on: Sat, 24 Jan 2015 04:59:37 +0000
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