LIMITS AND CONTINUITY

Question

Let f be the function defined as follows:

F(x) ={ |x-1|+2, for x<1
.           ax2+bx, for 1≤x<2 where a and b are constants
.           5x-10, for x≥2

 

Find a relationship between a and b for which f is continuous at x = 1.
Hint: A relationship between a and b just means an equation in a and b.

Answer 1

since f is cts ⇒ lim x→1+ (f(x)) =lim x→1^-  (f(x)) =f(1)

f(1) =a + b

lim x→1- (f(x))=2

⇒ a+ b = 2

also 

 

since f is cts ⇒ lim x→2+ (f(x)) =lim x→2- (f(x)) =f(2)

 

f(2) =0

and lim x→2- (f(x))= 4a + 2b

 

⇒ 4a + 2b = 0

 

so with the 2 eq  a+    b = 2

and                  4a + 2b= 0

⇒b = 4 and a = -2