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Question
In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.
`lim_(x -> 2) (x^2 - 1)` = 3
Solution
Here f(x) = x2 – 1, a = 2, I = 3
Let ∈ > 0 be given
Consider |f(x) – l| < ∈
∴ |(x2 – 1) – 3| < ∈
∴ |x2 – 4| < ∈
∴ |(x + 2)(x – 2)| < ∈
∴ |x + 2l lx – 2| < ∈
∴ |x – 2| < ∈ ...[∵ |x + 2| > 1]
∴ if we take δ = ∈, then
0 < |x – 2| < δ ⇒ |f(x) – 3| < ∈.
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