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Question
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= (|x + 1|)/(2x^2 + x - 1)",", "for" x ≠ -1),(= 0",", "for" x = -1):}}` at x = – 1
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Solution
|x + 1| `{:(= x + 1, ";" x ≥ -1),(= - (x + 1), ";" x < - 1):}`
∴ f(x) `{:(= (-(x + 1))/(2x^2 + x - 1), ";" x < -1),(= 0, ";" x = -1),(=(x + 1)/(2x^2 + x - 1), ";" x > - 1):}`
f(–1) = 0
`lim_(x -> 1^-) "f"(x) = lim_(x -> -1^-) (-(x + 1))/(2x^2 + x - 1)`
= `lim_(x -> -1^-) (-(x + 1))/((x + 1)(2x - 1))`
= `lim_(x -> -1^-) (-1)/(2x - 1) ...[(because x -> -1"," therefore x ≠ -1),(therefore x + 1 ≠ 0)]`
= `(-1)/(2(-1) - 1)`
= `1/3`
`lim_(x -> 1^+) "f"(x) = lim_(x -> -1^+) (x + 1)/(2x^2 + x - 1)`
= `lim_(x -> -1^+) (x + 1)/((x + 1)(2x - 1))`
= `lim_(x -> -1^+) (-1)/(2x - 1) ...[("As" x -> -1"," x ≠ -1),(therefore x + 1 ≠ 0)]`
= `1/(2(-1) - 1)`
= `(-1)/3`
∴ `lim_(x -> -1^-) "f"(x) ≠ lim_(x -> -1^+) "f"(x)`
∴ f(x) is discontinuous at x = – 1
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