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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) = {{:(2x + 1",", "if" x ≤ - 1),(3x",", "if" - 1 < x < 1),(2x - 1",", "if" x ≥ 1):}`
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Solution
`lim_(x -> 1^-) f(x) = lim_(x -> 1^-) 3x`
= 3 × 1
= 3
`lim_(x -> 1^+) f(x) = lim_(x -> 1^+) (2x - 1)`
= 2 × 1 – 1
= 2 – 1
= 1
∴ `lim_(x -> 1^-) f(x) ≠ lim_(x -> 1^+) f(x)`
Hence, `lim_(x -> 1) f(x)` does not exist.
∴ f(x) is not continuous at x = 1.
Let y = `f(x)`
| x | – 2 | – 1 | 0 | 1 | 2 | 3 |
| y |
2x + 1 – 3 |
2x + 1 – 1 |
3x 0 |
2x – 1 1 |
2x – 1 3 |
2x – 1 5 |
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