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.

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