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Sec. 1.2: Finding Limits Graphically and Numerically

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An Introduction to Limits Ex: What is the value of as x gets close to 2? Undefined ???

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Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex: x1.91.991.99922.0012.012.1 f (x) 2 12 2 11.4111.9411.994 undefined 12.006 12.06 12.61

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Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex:

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Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits (informal) Definition: Limit If f (x) becomes arbitrarily close to a single number L as x approaches c from both the left and the right, the limit as x approaches c is L.

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Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex:

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Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex: 1

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Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex: In order for a limit to exist, it must approach a single number L from both sides. In order for this limit to exist, the limit from the right of 2 and the limit from the left of 2 has to equal the same real number (or height). DNE

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An Introduction to Limits Ex: It would appear that the answer is – but this limit DNE because – is not a unique number. Sec. 1.2: Finding Limits Graphically and Numerically DNE

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An Introduction to Limits Ex: Sec. 1.2: Finding Limits Graphically and Numerically DNE ZOOM IN

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One-Sided Limits: Height of the curve approach x = c from the RIGHT Height of the curve approach x = c from the LEFT Definition (informal) Limit If the function f (x) becomes arbitrarily close to a single number L (a y-value) as x approaches c from either side, then the limit of f (x) as x approaches c is L written as * A limit is looking for the height of a curve at some x = c. * L must be a fixed, finite number.

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Definition (informal) of Limit: If then (Again, L must be a fixed, finite number.)

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Right and Left Limits To take the right limit, we’ll use the notation, The + symbol to the right of the number refers to taking the limit from values larger than 2. To take the left limit, we’ll use the notation, The – symbol to the right of the number refers to taking the limit from values smaller than 2.

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Limits can be estimated three ways: Numerically… looking at a table of values Graphically…. using a graph Analytically… using algebra OR calculus (covered next section)

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A limit does not exist when: 1. f(x) approaches a different number from the right side of c than it approaches from the left side. (case 1 example) 2. f(x) increases or decreases without bound as x approaches c. (The function goes to +/- infinity as x c : case 2 example) 3. f(x) oscillates between two fixed values as x approaches c. (case 3, example 5 in text: page 51)

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Section 1.2 – Finding Limits Graphically and Numerically

Section 1.2 – Finding Limits Graphically and Numerically

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