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Question
if `f(x) = { (mx^2 + n, x < 0),(nx + m, 0<= x <= 1),(nx^3 + m, x > 1):}`
For what integers m and n does `lim_(x-> 0) f(x)` and `lim_(x -> 1) f(x)` exist?
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Solution
(i) At x = 0,
`lim_(x → 0^-) f(x) = lim_(x → 0^-) (mx^2 + n) = n`
`lim_(x → 0^+) f(x) = lim_(x → 0^+) (nx + m) = m`
⇒ m = n
(ii) At x = 1,
`lim_(x → 1^-) f(x) = lim_(x → 1^-) (nx + m) = n + m = 2m, m ∈ R`
`lim_(x → 1^+) f(x) = lim_(x → 1^+) (nx^3 + m) = n + m = 2m, m ∈ R`
∴ m = n, for n ∈ R
`lim_(x → 0) f(x) = m, m ∈ R`
`lim_(x → 1) f(x) = 2m, m ∈ R`
Hence, for `lim_(x → 0) f(x)` to exist, m = n must be there; `lim_(x → 1) f(x)` exists for any integer values of m and n.
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