Find a value for the constant c so that the function is continuous at 7.

Question

f(x) = { c ÷ x-3 if x < 7

{ 5x-1 if x ≥ 7

c =

Is f then continuous at every real number? Yes or no? 

Answer 1

To find the value of c to make f(x), need to make both equations equal at x = 7:

c/(7-3) = 5(7)-1

c/4 = 34

c = 4(34) = 136

f(x) is not continuous at x = 3 which is a vertical asymptote.

So f(x) is not continuous at every real number. No.

Answer 2

The function is continuous if there are no holes or jumps.

For it to be continuous at 7 lim x-> 7+ f(x) = lim x-> 7- f(x) = f(7)

To the left of 7, we have the function c/(x-3).

So lim x->7- f(x) = lim x-> 7- c/(x-3)

To the right of 7, we have the function 5x - 1

So lim x-> 7+ f(x) = lim x-> 7+ 5x - 1