09Alimits

= Introduction to Limits =


 * A ** limit ** is a number that a function approaches (or gets closer and closer to)
 * in some circumstances, the function will eventually equal the limit
 * in other circumstances, the function will get closer and closer but may not ever reach the limit


 * Consider the function .. f(x) = x + 3

... ...
 * As x gets closer to 3, we can see that the value of f(x) will approach the value of 6.
 * In mathematical terms, we would say:
 * the limit of f(x) as x approaches 3 is 6


 * In notation, we would write it like this:

math . \qquad \qquad \begin{matrix} \lim \\ x \to 3 \\ \end{matrix} \;\; (x + 3) = 6 math


 * For any __**continuous**__ function, we can say that

math . \qquad \qquad \begin{matrix} \lim \\ x \to a \\ \end{matrix} \;\; f(x) = f(a) math


 * Convergent Series **


 * Consider the function

math . \qquad \qquad f(n) = \dfrac{4}{1} - \dfrac{4}{3} + \dfrac{4}{5} - \dfrac{4}{7} + \dfrac{4}{9} - \dfrac{4}{11} + \; ... \\.\\ . \qquad \qquad \qquad \text{where n is the number of terms being calculated, } n \in Z^+ \quad \{ \text{ Positive Integers} \} math


 * We can calculate a few values
 * I've rounded off to 6 decimal places but we can be as accurate as we choose:

... ... ... f(1) = 4 ... ... ... f(2) = 2.666667 ... ... ... f(3) = 3.466667 ... ... ... f(5) = 2.895238 ... ... ... f(6) = 3.339683 ... ... ... f(7) = 2.976046 ... ... ... f(8) = 3.283738 ... ... ... f(9) = 3.017072 ... ... .. f(10) = 3.252366


 * You can see that the values are gradually closing in on some value between 3 and 3.2
 * In fact, for this function, if we calculate an infinite number of terms, we would get p


 * In notation:

math . \qquad \qquad \begin{matrix} \lim \\ n \to \infty \\ \end{matrix} \;\; f(n) = \pi math


 * Calculating Limits **


 * To calculate a limit for a continuous function,
 * simply substitute the x value into the function


 * Example **

math . \qquad \text{Find } \begin{matrix} \lim \\ x \to 5 \\ \end{matrix} \;\; 2x^2 math


 * Solution:**

math . \qquad \begin{matrix} \lim \\ x \to 5 \\ \end{matrix} \;\; 2x^2 = 2(5)^2 math

math . \qquad \begin{matrix} \lim \\ x \to 5 \\ \end{matrix} \;\; 2x^2 = 50 math

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