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Question
At the given point x0 discover whether the given function is continuous or discontinuous citing the reasons for your answer:
x0 = 1, `f(x) = {{:((x^2 - 1)/(x - 1)",", x ≠ 1),(2",", x = 1):}`
Solution
`lim_(x -> 1^-) f(x) = lim_(x -> 1^-) (x^2 - 1)/(x - 1)`
= `lim_(x -> 1^-) ((x - 1)(x + 1))/((x - 1))`
= `lim_(x -> 1^-) (x + 1)`
= 1 + 1
= 2
`lim_(x -> 1^+) f(x) = lim_(x -> 1^+) (x^2 - 1)/(x - 1)`
= `lim_(x -> 1^+) ((x - 1)(x + 1))/((x - 1))`
= `lim_(x -> 1^+) (x + 1)`
= 1 + 1
= 2
`lim_(x -> 1^-) f(x) = lim_(x -> 1^+) f(x)` = 2
Hence `lim_(x -> 1) f(x)` = 2 ........(1)
`f(1)` = 2 ........(2)
From equation (1) and (2)
`lim_(x -> 1) f(x) = f(1)`
∴ f(x) is continuous at x0 = 1
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