Maths Calculus Derivatives & Limits

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Using the definition of the derivative evaluate the following limits

1) Lim h---> 0

[ ( 8 + h )^1/3 - 2 ] / h

2) Lim x ---> pi/3

( 2cosx - 1 ) / ( 3x - pi)

  • Maths Calculus Derivatives & Limits -

    Drag out L'Hospital's Rule

    1) 1/3 (8+h)^(-2/3) / 1
    = 1/3 * 1/4 = 1/12

    2) -2sinx/3 = -√3/3

  • Maths Calculus Derivatives & Limits -

    Oops. Using definition of derivative.
    Check back later. Lots of messy algebra.

  • Maths Calculus Derivatives & Limits -

    Use the binomial theorem to expand (8+h)1/3

    (8+h)1/3 - 2 = 81/3 + 1/3 8-2/3h - 1/3 * 2/3 8-5/3h2 + ...
    = -2 + 2 + 1/3 * 1/4 h - 2/9 * 1/32 h2 + ...

    Divide by h and all the terms with h2 or higher go away, leaving only:

    1/3 8-2/3 = 1/12

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