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2.2 Limits

2.2 Limits. Definition of Limit: Suppose that f(x) becomes arbitrarily close to the number L ( f(x)  L ) as x approaches a ( x  a ). Then we say that the limit of f(x) as x approaches a is L and write. Note 1: The number a may be replaced by ∞. 1. Example 1:

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2.2 Limits

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  1. 2.2 Limits Definition of Limit: Suppose that f(x) becomes arbitrarily close to the number L( f(x)L ) as x approaches a ( xa ).Then we say that the limit of f(x) as x approaches a is L and write Note 1: The number a may be replaced by ∞ 1

  2. Example 1: Estimate the limit of the function as x approaches 2. 2

  3. In the previous example Note 2: In general, the limit has nothing to do with the value of the function at a. All we are claiming is that the function approachesL. Example 2: Find the limit Solution: The function under the limit is not defined at x=1, but this doesn't matter because the definition of limit says that we consider value of x that are close to a, but not equal to a. 3

  4. Provided that x ≠a, the function can be reduced as follows: Then 4

  5. If the value of the limit coincides with the value of the function, the function is called continuous at this point. Definition of Continuity: A function f is continuous at x=a if f is defined at a and Exercises: Determine if the functions in the above examples are continuous at the given points (points of limits) 5

  6. Example: Find the trigonometric limit Continuous at 0? 6

  7. One-sided limits Definitions: If f(x) becomes arbitrarily close to the number L as x approaches a from the left, then we say that L is the left-sided limit of f(x) as x approaches a and write If f(x) becomes arbitrarily close to the number L as x approaches a from the right, then we say that L is the right-sided limit of f(x) as x approaches a and write 7

  8. Example: Heaviside function Continuous at 0? 8

  9. Exercise 3: Find the limit Solution: We use the fact that 1/x approaches 0 as x increases (or approaches infinity).So, we divide both numerator and denominator by the highest power of x and then use the above limit of 1/x. 9

  10. Theorem: If and then A. B. C. D. 10

  11. Three types of limit calculation (see numbered exercises): • Limit of a continuous function: just substitute the number. • 0/0 limit: cancel the common factor that gives 0 in the numerator and denominator and then substitute the number. • ∞/∞ limit (ratio of polynomials at ∞): divide both numerator and denominator by the highest power of x and use the fact that the limit of the reciprocal function at ∞ is 0. 11

  12. Homework Section 2.2: 11,15,27,31,37,41,49,51. 12

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