Fsc Part 2 Mathematics (Complete Solutions)
Q1
Determine the left hand limit and right hand limit and then find limit of the following function
Determine the left hand limit and right hand limit and then find limit of the following function
when x → c
when x → c
| (i) f(x) = 2x2 + x - 5, c = 1 | (ii) f(x) =
|
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| (iii) f(x) = |x - 5|, c = 5 |
Solution:
(i) f(x) = 2x2 + x - 5, c = 1
| Left hand limit | = | Lim
x → 1-
|
f(x) |
| = | Lim
x → 1-
|
(2x2 + x - 5) | |
| = | 2(1)2 + 1 - 5 | ||
| = | 2 + 1 - 5 | ||
| = | -2 | ||
| Right hand limit | = | Lim
x → 1+
|
f(x) |
| = | Lim
x → 1+
|
(2x2 + x - 5) | |
| = | 2(1)2 + 1 - 5 | ||
| = | 2 + 1 - 5 | ||
| = | -2 | ||
| Hence | Lim
x → 1-
|
f(x) | = | Lim
x → 1+
|
f(x) | = | -2 |
| ∴ | Lim
x → 1
|
ƒ(x) | = | -2 |
(ii) f(x) =
, c = -3
| x2 - 9 |
| x - 3 |
| Left hand limit | = | Lim
x → -3-
|
f(x) | ||
| = | Lim
x → -3-
|
|
|||
| = |
|
||||
| = |
|
||||
| = |
|
||||
| = | 0 |
| Right hand limit | = | Lim
x → -3+
|
f(x) | ||
| = | Lim
x → -3+
|
|
|||
| = |
|
||||
| = |
|
||||
| = |
|
||||
| = | 0 |
| Hence | Lim
x → -3-
|
f(x) | = | Lim
x → -3+
|
f(x) | = | 0 |
| ∴ | Lim
x → -3
|
ƒ(x) | = | 0 |
(iii) f(x) = |x - 5|, c = 5
| Left hand limit | = | Lim
x → 5-
|
f(x) |
| = | Lim
x → 5-
|
|x - 5| | |
| = | |5 - 5| | ||
| = | |0| | ||
| = | 0 | ||
| Right hand limit | = | Lim
x → 5+
|
f(x) |
| = | Lim
x → 5+
|
|x - 5| | |
| = | |5 - 5| | ||
| = | |0| | ||
| = | 0 | ||
| Hence | Lim
x → 5-
|
f(x) | = | Lim
x → 5+
|
f(x) | = | 0 |
| ∴ | Lim
x → 5
|
ƒ(x) | = | 0 |