|
Codification of the Limit ConceptWe now take a leap and formalized or codify the decimal pattern in the above examples.
Definition of Limit of a Function
|
|
Here d (delta) and e (epsilon) are Greek Letters. Use or think of letters D and E in place of them if you like.
In the latter case, a limit L is said to exist and we write
|
|
Real Player Videos
We write lim x® c g(x) = oo and say the value of the limits is oo when and only when for every whole number M > 0, there is a whole number k such that
g(x) > M whenever |x - c| < 10-k
And we write lim x® c g(x) = -oo and say the value of the limits is oo when and only when for every whole number M > 0, there is a whole number k such that
g(x) < - M whenever |x - c| < 10-k
When we say or write that a limit lim x® c g(x) has an infinite value (or approaches plus or minus infinity), we are describing the behavior of g(x) as x® c but we are not giving a finite number L as the limit. Thus a finite limit does not exist.
Remark (Technical Trap):
Now in speaking of limits, mathematics follows the technical convention that a limit
lim x® c g(x)
exists when and only when there is a real number L such that
So now we have a strange convention - blame human origins for it: The concept of a limit having an infinite value is defined, but a limit is said to exist when and only when the limit in question has a finite value.L = lim x® c g(x) .
Odd conventions like this in calculus provides a test of precision reading skills in calculus.
www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
Calculus Videos 0. First Calculus Preview 0. Triangle Inequality 0 Inequalities 0. Solving Inequalities 1. Distance+Midpt Formulas 1. Function Domains 1.Polynomial Domain+Range 1. Fn: Linear Combinations 1. Fn Composition II 1. Fn Composition I 1. Solving y**n = x**m 2 .Real Numbers 2. Limits Numerical View 2. Limits,. Formal Definition 2. Limit Properties Numerically 2. Decimal View of Limits 2. Error Control View 2. Limits & Continuity 2. Limit Vals via Substitution 2. Limits & Composite Fns 2.. Limit Examples 2. One Sided Limits 2. Infinity and Limits 2. Parameters in Limits 3. Derivative Motivation 3. Derivative Definition I 3. Derivatives Definition II 3. Calculus: Why Radians 3 d/dx of sin(x) & cos(x) (I) 3 d/dx of sin(x) & cos(x) (II) 3.Sum Rule 3. Product Rule 3. Power Rule 3. Previous Rules Combined 3. d/dx for Polynomials 3. Reciprocal Rule 3. Reciprocal Law: sec & csc 3. Reciprocals & Power Rule 3. Power Law for Integers < 0 3. Quotient Rule 3. Quotient Rule Examples 3. Quotient Rule: tan & cot 3. Linear Chain Rule 3. Chain Rule for Powers 3. Chain Rule - Polynomials 3. Chain Rule Examples I 3. Chain Rule Examples II 3. Linear Approximation I 3. General Chain Rule 3. Inverse Fns Derivatives 3. Chain Rule: ln(x) & exp(x) 3. Square & Cube Roots 4. Linear Approximation 4. Second Derivative Test 4. Sketch y = x^3 - 6x^2- 12x 4. Sketch y = x^3 - 3 x^2 - 9x 4. Sketch y = 1 - 1/(1+x^2) 5. Indefinite Integrals A 5. Indefinite Integrals B. 5 Indefinite Integrals C 6. Definite Integral D 6. Area Under Curves 7. Volume of a Sphere To Learn More, visit Volumes 2 and 3.
Advanced Topics
Limit Properties Algebraically Pigeon Hole Principle Bolzano Constant Difference Thm Continuous Functions Rational Functions Mean Value Theorem One Side Range Theorem Range On One Side From Lipschitz Continuity
To Learn More: Visit Real Analysis.
|
www.whyslopes.com
|