Find the value of the constant c that makes the following function continuous on (-infinity,infinity).
F(x)= cx+3 if x ∈ (-infinity, 6]
F(x)= cx^2 -3 if x ∈ (6, infinity)

Question

Find the value of the constant c that makes the following function continuous on (-infinity,infinity). 
F(x)= cx+3 if x ∈ (-infinity, 6] 
F(x)= cx^2 -3 if x ∈ (6, infinity)

abb0t · Answer

TAKE THE LIMIT frmo the left and right of 6

hartnn · Answer

for F(x) to be continuous everywhere, it has to be continuous at x=6 , right ?

so, just equate cx+3 with cx^2-3 
and solve the quadratic to get 2 values of c

anonymous · Answer

can we  equate them why to solve the  two equation??

hartnn · Answer

you can equate them because, for continuity, left hand limit should equal right hand limit

so, lim x->6 cx+3 = lim x->6 cx^2-3 
from this c can be found easily :)

anonymous · Answer

yes   c  can be found  out  very  easily

anonymous · Answer

Can you show me how because i am not understanding

anonymous · Answer

hartnn just  wrote above  it   just  put the  limit of  x as  6 in both the equation  equate then  c  will come out  to b  1/5

hartnn · Answer

which part exactly, you are not getting ?

anonymous · Answer

when you say put the limit of x as 6 does that mean put 6 in for x?

hartnn · Answer

absolutely

hartnn · Answer

lim x->6 cx+3 = lim x->6 cx^2-3 

so plug in x=6 in 
cx+3 = cx^2 -3

anonymous · Answer

so then you get c9 and c33. how do you get 1/5?

hartnn · Answer

its not c(x+3) its cx +3

when you put x=6
you get 
6c+3

similarly do it for right side

anonymous · Answer

okay so the other side would be 36c-3 right? and you set them equal to each other?

hartnn · Answer

yes, correct, go on :)

anonymous · Answer

I keep getting c= 5

hartnn · Answer

6c+3 = 36c-3
30 c = 6 
c= ....?

anonymous · Answer

oh okay i see. Thank you so much

hartnn · Answer

welcome ^_^