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Properties of Limits
Numerical examples below with limits involving sums, products, multiples and quotients of functions suggest the algebraically described patterns or properties of limits
Theorem E.2 [ Properties of Limits] Assume
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The demonstrations (university level) depend on inequalities. Skip them for now . All the demonstrations, depend on error control estimates for calculations. To be more precise, we should say that f(x) and g(x) above should be real valued functions defined on a-deleted interval centered about x = a Here the phrase a-deleted to signal an about about the point x = a with the value a removed.
Most students are content to see and understand what the properties say and look no further. Students want to see the properties in calculus proven should have been patience. Learn the numerical and graphical patterns that suggest and illustrate them, and later, after you have develop and practiced your algebraic and logic skills with the properties, you can study the proofs. They are available. Our aim here to provide you with an operational command of calculus and in doing so tell you where more could said (as is in proofs). The more can be left for later. An operational command in the first instance can come from examples and the statements of properties or theorems you need to know and apply.
My advice here about postponing or skipping the proofs is contrary to what I wanted. As a student, I wanted to see the logic in full, the justification of formulas and methods, before I would use them. That slowed my learning and could have led to failure. Full justification is best left for later.
The following numerical examples could have been given first to suggest the above properties.
Here is a sequence of numbers
x
...
Observed the above sequence of x-values approach a = 1.54
Two Functions
Let
Initial Limits
Numerical calculation gives the following (Ignore the subscript k below)
Therefore as x approached a = 1.54 through the x-values above: f(x) and g(x)
respectively approach 
Limits of Sums and Products
Numerical Calculation gives
So we see when x approaches a = 1.54 the sum g(x) + f(x)
approaches 
and the product f(x)g(x)
approaches 
Limits of Differences, Quotients and Constant Multiples
Let c = 4. Then Numerical calculation gives
Therefore as x approaches a = 1.54, we have
f(x)-g(x) --->
cf(x) --->
f(x)
g(x)--->
Reciprocals of a function
Numerical calculation gives
The foregoing suggest
| 1 g(x) |
---> |  |
Again, the above examples suggest a pattern or two which can be algebraically described. They agree with and suggest the following algebraically described properties of limits.
Theorem E.2 [ Properties of Limits] Assume
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The demonstrations are omitted. All the demonstrations, depend on error control estimates for calculations.
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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.