Posted by **Yousef** on Tuesday, October 25, 2011 at 9:19am.

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 -
**Steve**, Tuesday, October 25, 2011 at 10:25am
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 -
**Steve**, Tuesday, October 25, 2011 at 10:26am
Oops. Using definition of derivative.

Check back later. Lots of messy algebra.

- Maths Calculus Derivatives & Limits -
**Steve**, Tuesday, October 25, 2011 at 10:48am
Use the binomial theorem to expand (8+h)^{1/3}

(8+h)^{1/3} - 2 = 8^{1/3} + 1/3 8^{-2/3}h - 1/3 * 2/3 8^{-5/3}h^{2} + ...

= -2 + 2 + 1/3 * 1/4 h - 2/9 * 1/32 h^{2} + ...

Divide by h and all the terms with h^{2} or higher go away, leaving only:

1/3 8^{-2/3} = 1/12

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