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. Blast from the Past Find point(s) of intersection 1. 2. 3.4. 5.

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. Blast from the Past Find point(s) of intersection 1. 2. 3.4. 5.

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. Blast from the Past In exercises 1 – 4 write the inequality in the form 1. 2. 3. 4. In exercises 5 and 6 write the fraction in reduced form 5. 6.

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. Blast from the Past In exercises 1 – 4 write the inequality in the form 1. 2. 3. 4. In exercises 5 and 6 write the fraction in reduced form 5. 6.

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Finding Limits Graphically and Numerically Limits are restrictions…mathematically f(x) is restricted by the graph of the function. In this section you will estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist.

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Lets analyze the graph of f (x):. To get an idea of the behavior of the graph of f (x) near x = 0 lets use a set of values approaching 0 from the left and set of values approaching 0 from the right. Estimating a Limit Numerically

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Slide 2- 7 Analyzing the table we see although x cannot equal 0,we can come arbitrarily close to 0 and as a result the function f(x) moves arbitrarily close to 2. x-0.01-0.001-0.0001-0.0000100.000010.00010.0010.01 f(x)1.99491.999501.99995 2.000052.00052.004992.0049 Which leads to the limit notation: Which reads as “the limit of f(x) as x approaches 0 is 2

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. Exploration: Estimate numerically the x0.750.90.990.99911.0011.011.11.25 f(x)2.3132.7102.9702.997?3.0033.0303.3103.813 X1.751.91.991.99922.0012.012.12.25 f(x)????????? Complete the chart and estimate numerically the

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Finding a Limit Graphically Use the graph of f(x) to determine each limit:

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Behavior That Differs from Right to Left: The following graphs of f(x) represent functions that the limit of f(x) as x, approaches a, does not exist. As x approaches a from the right: Limits That Fail to Exist As x approaches a from the left: Because f(x) approaches a different number from the left side of a than it approaches from the right side, the limit does not exist.

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Unbounded Behavior: The following graph of f(x) represent function that the does not exist Oscillating Behavior: The following graph of f(x) represent function that the does not exist

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Exit Ticket 1.Write a brief description of the meaning of the notation 2.If, can you conclude anything about the limit of f(x) as x approaches 2? Explain your reasoning.

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. Blast from the Past In exercises 1 and 2 rationalize the numerator 1. 2. In exercise 3 simplify expression 3. 4. If then

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. Blast from the Past In exercises 1 and 2 rationalize the numerator 1. 2. In exercise 3 simplify expression 3. 4. If then

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5. Use the graph to find the limit (if it exists). If the limit does not exist, explain why.

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. 6. Use the graph to find the limit (if it exists). If the limit does not exist, explain why.

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7. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not explain why. a. b. c. d.

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8. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not explain why. a. b. c. d. e. f.

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Evaluating Limits Analytically In this section you will evaluate a limit using properties of limits. Develop a strategy for finding limits. Evaluate a limit using dividing out and rationalizing techniques..

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Previous section we learned that does not depend on the value of the function at x = c. However, it may happen a limit may be precisely f(c), the limit can be evaluated by Direct Substitution. These well-behaved functions are continuous. Analytical Approach

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Determine the limits of the functions Remember a limit may be precisely f(c) and the limit can be evaluated by Direct Substitution if the function is continuous at x = c.

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Properties of Limits: Where f(x) and g(x) are functions with limits that exists such thatand 1. Scalar multiple: where b is a constant 2.Sum or difference: 3.Product: 4.Quotient: 5.Power:

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Indeterminate Form of a Limit Direct Substitution may produce the solution 0/0, an indeterminate form Dividing Technique Rationalizing Technique Simplifying Technique

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Exploration In exercises 1 – 5 evaluate the limits 1. 2. 3. 4. 5.

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Limits of Trigonometric Functions Let a be a real number in the domain of the given trigonometric function. Two Special Trigonometric Limits 1. 2.

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Exploration of Limits of Functions The principle of direct substitution can be expanded to include exponential, logarithmic, piece-wise, and inverse trigonometric functions 1. 5. 2.6. 3.7. 4.8.

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. Exit Ticket 1.Letfind 2.Let find 3.What is meant by indeterminate form?

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