Asked by Yousef

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)
Answered by Steve
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
Answered by Steve
Oops. Using definition of derivative.
Check back later. Lots of messy algebra.
Answered by Steve
Use the binomial theorem to expand (8+h)<sup>1/3</sup>

(8+h)<sup>1/3</sup> - 2 = 8<sup>1/3</sup> + 1/3 8<sup>-2/3</sup>h - 1/3 * 2/3 8<sup>-5/3</sup>h<sup>2</sup> + ...
= -2 + 2 + 1/3 * 1/4 h - 2/9 * 1/32 h<sup>2</sup> + ...

Divide by h and all the terms with h<sup>2</sup> or higher go away, leaving only:

1/3 8<sup>-2/3</sup> = 1/12
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