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

Answer 1

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| < ∈.