Fsc Part 2 Mathematics (Complete Solutions)

# Q2 Discuss thecontinuity of f(x) at x = c

 (i)    f(x)    = { 2x + 5    if x ≤ 2 4x + 1    if x > 2

Solution:

We have to find the countinuty of f(x) at x = 2

(a)    f(2) = 2(2) + 5
f(2) = 4 + 5
f(2) = 9     ------------------------------------ (1)

 (b) Lim x → 2 f(x) = ?
 Left hand limit = Lim x → 2- f(x) = Lim x → 2- (2x + 5) = 2(2) + 5 = 4 + 5 = 9

 Right hand limit = Lim x → 2+ f(x) = Lim x → 2+ (4x + 1) = 4(2) + 1 = 8 + 1 = 9

 Hence Lim x → 2- f(x) = Lim x → 2+ f(x) = 9

 ∴ Lim x → 2 ƒ(x) = 9     ------------------------------------ (2)

 From equation    1 and    2    we get Lim x → 2 ƒ(x) = f(2)
∴      f(x) is continuous function at      x = 2

(ii)    f(x)    =   {
 3x - 1 if x < 1 4 if x = 1   ,  c = 1 2x if x > 1

Solution:

We have to find the countinuty of f(x) at c = 1

(a)    f(1) = 4

f(1) = 4     ------------------------------------ (1)

 (b) Lim x → 1 f(x) = ?
 Left hand limit = Lim x → 1- f(x) = Lim x → 1- (3x - 1) = 3(1) - 1 = 3 - 1 = 2

 Right hand limit = Lim x → 1+ f(x) = Lim x → 1+ (2x) = 2(1) = 2

 Hence Lim x → 1- f(x) = Lim x → 1+ f(x) = 2

 ∴ Lim x → 1 ƒ(x) = 2     ------------------------------------ (2)

 From equation    1 and    2    we get Lim x → 1 ƒ(x) ≠ ƒ(1)
∴      ƒ(x) is discontinuous.