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Question
Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.
`f(x) = {{:((x - 1)^3",", "if" x < 0),((x + 1)^3",", "if" x ≥ 0):}`
Solution
`f(x) = {{:((x - 1)^3",", "if" x < 0),((x + 1)^3",", "if" x ≥ 0):}`
`lim_(x -> 0^-) f(x) = lim_(x -> 0^-) (x - 1)^3`
= (0 – 1)3
= – 1 .........(1)
`lim_(x -> 0^+) f(x) = lim_(x -> 0^+) (x + 1)^3`
= (0 + 1)3
= 1 .........(2)
From equation (1) and (2) we have
`lim_(x -> 0^-) f(x) ≠ lim_(x -> 0^+) f(x)`
∴ `lim_(x -> 0) f(x)` does not exist.
Hence f(x) is not continuous at x = 0.
| x | – 2 | – 2 | 0 | 1 | 2 |
| y |
(x – 1)3 – 8 |
(x – 1)3 – 27 |
(x + 1)3 1 |
(x + 1)3 8 |
(x + 1)3 27 |
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