Unit One Notes Limits, Models, and Fundamentals

Unit One Notes Limits, Models, and Fundamentals - Unit ONE...

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Unformatted text preview: Unit ONE NOTES—Fundamentals plus Introduction to Change and Models Four Perspectives/Representations/Viewpoints of Functions: 1. Algebraically 2. Graphically 3. Numerically 4. Verbally Ch 1—Introduction to Limits The concept of a limit is the basic idea that leads to many applications in Calculus. What we will be investigating is what happens to the values of f(x)_______ of a function of f as the variable x approaches the number a . Example 1: Given f(x) = x x 2 2-- x 1.98 1.99 2 2.01 2.02 f(x) Formally, the D limit of a function_________________________________ ____________________________________________________________. and_________________________________________________________ In words this means ___________________________________________ ____________________________________________________________ For a limit to exist, the following must be true: ______________________ Evaluate the limits below using a graphical and numeric approach: Ex 2. ( 29 4 lim 2 2-- x x Ex 3. --- 3 5 2 lim 3 x x x Ex 4. x x x + ∞- 1 1 lim Ex 5. ( 29 x f x 2 lim- if f(x) ( 29 - <- = 2 , 4 2 , 1 2 2 x x x x x f Ex 6. - 2 1 lim x x Finding Limits Using an Algebraic Approach: D ___________________________________________________________________ Ex 1. ( 29 6 3 4 lim 2 3 2 +-- x x x Ex 2. --- 2 4 lim 2 2 x x x Ex 3. ---- 3 6 lim 2 3 x x x x Ex 4. - +- h h h 6 36 lim Ex 5. --- 1 1 lim 1 x x x Ex 6. ( 29 -- 2 1 1 1 lim x x Ex 7. -- 2 5 lim 2 x x Evaluating Infinite Limits and End Behaviors Infinite Limit Guidelines for Rational Functions: 1. If the degree of the numerator is greater than the degree of the denominator, then limit f(x) is ∞ (dne) or - ∞ (dne)—dependent upon the end behaviors. 2. if the degree of the numerator is equal the degree of the denominator, then limit f(x) = ratio of the lead coefficients. 3. If the degree of the numerator is less than the degree of the denominator, then limit f(x) is 0. Ex 1: a) +-- ∞- 3 6 2 5 lim 2 x x x x b) +-- ∞-- 3 6 2 5 lim 2 x x x x Ex 2: - + ∞- 9 5 5 3 lim x x x Ex 3: --- ∞- 10 7 10 8 lim 2 x x x x Ex 4: -- ∞- x x e 9 . 2 1 8 lim DOT/CROSS TECHNIQUE The dot/cross technique is a quick method for solving polynomial P(x) and rational R(x) inequalities of the form P(x) > 0, P(x) > 0, P(x) < 0, P(x) < 0 or R(x). While it is not necessary to use the dot cross technique to solve a linear inequality, the method is based upon a very simple idea involving real linear factors....
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Unit One Notes Limits, Models, and Fundamentals - Unit ONE...

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