Posted by Yousef on .
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,
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,
Oops. Using definition of derivative.
Check back later. Lots of messy algebra. 
Maths Calculus Derivatives & Limits 
Steve,
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