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Question
Evaluate the following :
`lim_(x -> 0)[x/(|x| + x^2)]`
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Solution
We know that |x| = x if x > 0
= – x if x < 0
∴ `lim_(x -> 0^+) [x/(|x| + x^2)]`
= `lim_(x -> 0) x/(x + x^2)`
= `lim_(x -> 0) x/(x(1 + x))`
= `lim_(x -> 0) 1/(1 + x)` ...[∵ x → 0, ∴ x ≠ 0]
= `(lim_(x -> 0) 1)/(lim_(x -> 0) (1 + x))`
= `1/(1 + 0)`
= 1
`lim_(x -> 0^-) [x/(|x| + x^2)]`
= `lim_(x -> 0) x/(-x + x^2)`
= `lim_(x -> 0) x/(x(-1 + x))`
= `lim_(x -> 0) 1/(-1 + x)` ...[∵ x → 0, ∴ x ≠ 0]
= `(lim_(x -> 0) 1)/(lim_(x -> 0) (-1 + x))`
= `1/(-1 + 0)`
= – 1
∴ `lim_(x -> 0^+) [x/(|x| + x^2)] ≠ lim_(x -> 0^-) [x/(|x| + x^2)] `
∴ `lim_(x -> 0) [x/(|x| + x^2)]` does not exist.
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