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
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)
3 answers
Oops. Using definition of derivative.
Check back later. Lots of messy algebra.
Check back later. Lots of messy algebra.
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
(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