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5 2 limit-ofa_function

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  • 1. Limit of a Function
  • 2.
    • The concept of a limit is central to the study of calculus.
    • Differentiation and integration are each based on the notion of a limit.
    • Limit of f ( x ) as x approaches a – the value that f ( x ) approaches as x gets closer to a
    • In symbols:
  • 3.
    • Observe the values of f ( x ) = x 2 + 1 as x approaches 3:
    • We say that .
    x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4 f ( x ) 5 7.25 9.41 9.94 9.994 10 10.006 10.06 10.61 13.25 17
  • 4.  
  • 5.
    • Given . Find the limit of f ( x ) as x approaches 3. .
    • We still say that .
    x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4 f ( x ) 5 7.25 9.41 9.94 9.994 15 10.006 10.06 10.61 13.25 17
  • 6.
    • Given . Find the limit of f ( x ) as x approaches 2. .
    x 1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3 f ( x ) 3 4 4.8 4.98 4.998 4 4.004 4.04 4.41 6.25 9
  • 7.
    • The limit of the function f exists only if the left-hand and right-hand limits both exist and are equal.
    • exists if and only if
  • 8. Limit Theorems
  • 9.
    • L1. The limit of a constant function is the constant.
    • Examples.
  • 10.
    • L2. If f is a polynomial function, then
    • Example.
  • 11.
    • L3. The limit of a sum is the sum of limits.
    • Example.
  • 12.
    • L4. The limit of a constant times a function is the constant times the limit of the function.
    • Example.
  • 13.
    • L5. The limit of a product is the product of the limits.
    • Example.
  • 14.
    • L6. The limit of a quotient is the quotient of the limits.
    • Example.
  • 15.
    • L7. The limit of a function raised to an exponent is the limit of the function raised to that exponent.
    • Example.
  • 16.
    • Example. Find .
  • 17.
    • Example. Find .
  • 18.
    • Example. Given Find . .
  • 19.
    • 1.a Find .
  • 20.
    • 1.d Find .
  • 21.
    • 2.c Find .
  • 22.
    • 3.a Find .
  • 23.
    • 3.c Find .
  • 24.
    • 4.b Find .
  • 25.
    • 5.a Suppose and . Find . .
  • 26.
    • 7. Given . Find . .

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