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Question
Discuss the continuity of the function f(x) = |2x + 3|, at x = `(−3)/(2)`
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Solution
f(x) = |2x + 3|, x = `(−3)/(2)`
|2x + 3| = 2x + 3; `x ≥ (-3)/(2)`
= – (2x + 3); `x < (-3)/(2)`
`lim_(x -> (-3^(-))/(2)) "f"(x) = lim_(x -> (-3^(-))/(2)) |2x + 3|`
= `lim_(x -> (-3^(-))/(2)) [- (2x + 3)]`
= `-[2((-3)/2) + 3]`
= 0
`lim_(x -> (-3^(+))/(2)) "f"(x) = lim_(x -> (-3^(+))/(2)) |2x + 3|`
= `lim_(x -> (-3^(+))/(2)) (2x + 3)`
= `2 ((-3)/2) + 3`
= 0.
`"f"((-3)/2) = |2 ((-3)/2) + 3|`
= |0|
= 0
∴ `lim_(x -> (-3^(-))/(2)) "f"(x) = lim_(x -> (-3^(+))/(2)) "f"(x) = "f"((-3)/2)`
∴ f(x) is continuous at x = `(-3)/(2)`.
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