Question

Dadas as funções f(x) = x2, f(x) = 2x2+3 e f(x) = 1/x, determine, para cada uma:
lim [ f(2 + z) – f(2) ] / z “.
z->0

Answer 1

lim   [ f(2 + z) – f(2) ] / z 
z->0 

f(x)=x²

lim   [ (2+z)² – 2² ] / z 
z->0 

lim   [2z+z²  ] / z 
z->0 

lim   [4+2z+z² – 4 ] / z 
z->0 

lim   2+z =2+0 =2 
z->0 

================================================
f(x) = 2x² +3

lim   [ f(2 + z) – f(2) ] / z 
z->0 

lim   [2*(2+z)+3– 2*2²+3] / z 
z->0 

lim   [2*(2+z)+3– 2*2²+3] / z 
z->0 

lim   [4+2z+3– 4+3] / z 
z->0 

lim   [2z] / z  = lim 2  =2
z->0               z-->0

#########################################

f(x) = 1/x

lim   [ f(2 + z) – f(2) ] / z 
z->0 

lim   [ 1/(2+z) – 1/2 ] / z 
z->0 

lim   [ (2-(2+z))/2(2+z) ] / z 
z->0 

lim   [ z/2(2+z) ] / z 
z->0 

lim   1/2(2+z)   =1/(4+0)=1/4
z->0