Limits-Numerical+and+Definition

The limit of a function is what that function's value becomes as its argument approaches a certain value. Limits are useful because for some arguments, the function itself is undefined, yet as the argument gets closer to the value which would make the function undefined, the function clearly approaches a certain value.
 * [Section 1] Introduction and the Numerical Approach:

Example #1 - What is the value of f(x) = 1/x as x approaches 0? f(0) is undefined, but values for x very close to 0 always yield an answer. Though 0.00000001 is very close to 0, if x=0.0000000001, the function will still be solvable. So to solve for the limit of f(x) as x goes to 0, solve for f(x) while keeping x arbitrarily close to 0. ** It appears that as x approaches 0 from the right, f(x) approaches infinity, so it may be said that " **the limit of f(x)=1/x as x approaches 0 from the right is infinity". ** Note that this function does not truly have a limit at x=0 because if values close to but below x=0 were tested, the function progresses to negative infinity. It appears that as x approaches 0 from the left, f(x) approaches negative infinity, so it may be said that " **the limit of f(x)=1/x as x approaches 0 from the left is negative infinity". ** This raises the critical observation that for a limit to exist, the function must approach the same value from both sides. This example is designed to illustrate our ability to continually regress x to smaller and smaller values away from that which would make the function undefined.
 * x || 1/x ||
 * 1 || 1 ||
 * 0.5 || 2 ||
 * 0.25 || 4 ||
 * 0.1 || 10 ||
 * 0.01 || 100 ||
 * 0.001 || 1,000 ||
 * 0.0001 || 10,000 ||
 * 0.000000001 || 100000000 ||
 * x || 1/x ||
 * 1 || 1 ||
 * 2 || 0.5 ||
 * 4 || 0.25 ||
 * 10 || 0.1 ||
 * 100 || 0.01 ||
 * 1,000 || 0.001 ||
 * 10,000 || 0.0001 ||
 * 100000000 || 0.000000001 ||

Conclusion: Although at certain values for x, a function f(x) may be undefined, a limit may still exist there, provided the function is "approaching" some value from both sides of the undefined point. We can get very close to the true limit of f(x) at such undefined arguments if we calculate the value of f(x) at values very close to, but not precisely, the value of the undefined argument.

Example #2 -

Example #3 - What is the value of **f(x)=(x ** **2 ** **-1)/(x-1) ** when x=1?

First, substitute "1" for "x" and see the result: (1 2 -1)/(1-1) = (1-1)/(1-1) = 0/0

**At x=1, f(x) = 0/0 and is "indeterminate", meaning its value can’t be determined (see the graph above). So instead of trying to solve for x=1, try solving for x as x approaches 1: ** The function must be tested <span style="font-family: Euclid,helvetica,sans-serif; font-size: 12px; line-height: normal;">**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12px; line-height: normal;">from both directions ** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12px; line-height: normal;"> to be sure f(x) approaches the same value of x…
 * x || (x 2 -1)/(x-1) ||
 * 1.5 || 2.5 ||
 * 1.1 || 2.1 ||
 * 1.01 || 2.01 ||
 * 1.001 || 2.001 ||
 * 1.0001 || 2.0001 ||
 * 1.00001 || 2.00001 ||
 * x || (x 2 -1)/(x-1) ||
 * 0.5 || 1.5 ||
 * 0.9 || 1.9 ||
 * 0.99 || 1.99 ||
 * 0.999 || 1.999 ||
 * 0.9999 || 1.9999 ||
 * 0.99999 || 1.99999 ||

<span style="font-family: Arial,Helvetica,sans-serif;"> It appears that as x approaches 1, f(x) approaches 2, so it may be said that " **<span style="font-family: Arial,Helvetica,sans-serif;">the limit of f(x)=(x2-1)/(x-1) as x approaches 1 is 2". **<span style="font-family: Arial,Helvetica,sans-serif;"> However, one **<span style="font-family: Arial,Helvetica,sans-serif;">cannot say what the value at x=1 really is, because at x=1, f(x) = 0/0, and 0/0 is a value that cannot be determined. **<span style="font-family: Arial,Helvetica,sans-serif;">

Numerical approaches to limits where the function is undefined are only capable of estimating heuristically the correct value that a function approaches at the specified argument. The formal definition algebraically determines the precise value...
 * [Section 2] Formal Definition:**

<span style="font-family: Arial,serif; font-size: 12px; line-height: normal;">** The limit of f(x) as x approaches c is L ** <span style="font-family: Arial,serif; font-size: 14.4px; line-height: normal;">if and only if, given <span style="font-family: Arial,serif; font-size: 12px; line-height: normal;">**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 16px; font-weight: normal;">ε > 0 ** <span style="font-family: Arial,serif; font-size: 14.4px; line-height: normal;">, there exists <span style="font-family: Arial,serif; font-size: 12px; line-height: normal;">** <span style="font-family: 'Times New Roman',serif; font-size: 16px; font-weight: normal; line-height: 19px;">δ > 0 ** <span style="font-family: Arial,serif; font-size: 14.4px; line-height: normal;"> such that <span style="font-family: Arial,serif; font-size: 12px; line-height: normal;">** 0 < |x - c| < ** <span style="font-family: Arial,serif; font-size: 14.4px; line-height: normal;"><span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;">δ implies that <span style="font-family: Arial,serif; font-size: 12px; line-height: normal;">** |f(x) - L| < ** <span style="font-family: Arial,serif; font-size: 14.4px; line-height: normal;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: 16px;">ε.

Verbally - The limit of f(x) as x approaches c is the number L such that f(x) can be any positive <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: 16px;">ε units from L, no matter how small ε is, given x is some positive number <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;">δ (delta) away from c. x cannot equal c, and so f(x) cannot equal L, for there are functions in which f(c) is undefined and never reaches L.

media type="custom" key="5548931" Reference Video

Example #1 - What is the limit of f(x) = (x-3) (1/3) + 2 as x approaches 3? At x=3, f(x) is undefined. Graphically or numerically, we can conclude the limit at x=3 likely equals 2. To confirm this formally, we rewrite the equation... (2+<span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">ε) = [(x+ <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;">δ) <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px; vertical-align: super;">(1/3) <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;">] + 2

δ <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px; vertical-align: super;">(1/3) <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;"> = <span style="font-family: Arial,Helvetica,sans-serif; font-size: 16px; line-height: normal;">ε <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;"> <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;">δ = ε <span style="font-family: arial,sans-serif; font-size: small; line-height: normal; vertical-align: super;">3 <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">

Therefore, as x approaches 3, there is always a value <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;">δ that can be added to x <span style="font-family: Arial,Helvetica,sans-serif; font-size: small; line-height: normal;">t <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">hat makes the difference between f(x) and L (also called ε) smaller.

Example #2 - What is the limit of f(x) = <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">x <span style="font-family: arial,sans-serif; font-size: small; line-height: normal; vertical-align: super;">2 <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">+2x-3 as x approaches 3?

Firstly, because the value for f(1) is defined, you can use direct substitution to find the limit as x approaches 1. However for these purposes, the use of the formal definition of a limit will be used.

According to the definition of a limit, there is an infinitely small (but still greater than 0) value of ε, in this example, use <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">ε <span style="font-family: 'Times New Roman',serif; font-size: 16px; line-height: 19px;"> = 0.01

there fore, we have the equation that -.001 < <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">x <span style="font-family: arial,sans-serif; font-size: small; line-height: normal; vertical-align: super;">2 <span style="font-family: arial,sans-serif; font-size: small; line-height: normal;">+2x-3 < 0.01 --> meaning that the value for limit will be within -0.01 and 0.01 using inequalities, solve the equation for x which becomes 0.9975 < x < 1.0025

thus the limit as x approaches 1 = 0.

__Works Cited__

http://archives.math.utk.edu/visual.calculus/1/definition.6/index.html __[]__ http://www.5min.com/Video/The-Formal-Definition-of-a-Limit-169078903 __[]__ <span style="font-family: Euclid,helvetica,sans-serif;">__[|http://archives.math.utk.edu/visual.calculus/1/definition.6/4.html]__

The book.