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Definition of Limit Lesson 2-2

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Verbal Definition of Limit L is the limit of f (x) as x approaches c if and only if L is the one number you can keep f (x)(x) arbitrarily close to just by keeping x close to c,c, but not equal to c.c.

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Example: A rational algebraic function Why is there no value for the function at x = -2? Yes! The denominator would go to zero. Then you would be dividing by zero. Yuck!!!!

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Explore f (x) at x = -2 Looking back at the graph, what number does f (x) appear to be at x = -2?

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Explore f (x) at x = -2 In TBLSET set tblStart… Δ…… Press ENTER. Then view the TABLE.x y undef Looking at the table, what number does f (x) appear to be at x = -2?

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Looking at both graph and table of f(x) we can conclude the limit as x approaches -2 is 2.x y undef How about finding the limit algebraically?

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Finding a limit algebraically… For the original equation, x at -2 does not exist. We say its approaching a value of 2.

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Removable Discontinuity The function is discontinuous because of the gap at x = -2 but the gap can be removed by defining f (-2) to be 2. An open circle at the point of discontinuity is used to illustrate that.

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Step Discontinuity This function is discontinuous because of the gap at x = 0. However, it can not simply be removed since there is a large “step” between the two branches. What is the limit at x = 2? Answer: -1 The discontinuity is at x = 0. As x approaches 2 from the left, g(x) is close to -1. As x approaches 2 from the right, g(x) is still close to -1. What is the limit at x = 0? Answer: Does not exist.

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Formal Definition of Limit: L is the limit of f (x) as x approaches c if and only if for any positive number epsilon (ε), no matter how small, there is a positive number delta (δ)(δ) such that if x is within delta units of c (but not equal to c)c) then f (x) is within epsilon units of L.L.

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Formal Definition of Limit L is the limit of f (x) as x approaches c if and only if for any positive number ε,ε, no matter how small, there is a positive number δ such that if x is within δ units of c (but not equal to c)c) then f (x) is within ε units of L L is 4 c 2 ε 1 “arbitrarily” δ is 0.8 and 0.6 Pick the δ that is most restrictive. So δ is 0.6

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What is the limit at x = 2? Answer: No limit What is the limit at x = 5? Answer: 2 The value of h(5) is 4 but h(x) approaches 2 from the left and from the right.

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Example: Given From the graph, what do you think the limit of f (x) is as x approaches 1? Try to evaluate f (1) by direct substitution. What form does the answer take? What is this form called?

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Factor the numerator and simplify the expression. Although the simplified expression does not equal f (1), you can substitute 1 for x and get an answer. What is the answer and what does it represent? Ω

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