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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)(2x + 3)` = 7
Solution
Here, f(x) = 2x + 3, a = 2, l = 7
Let ∈ > 0 be given.
Consider |f(x) – l| < ∈
∴ |(2x + 3) – 7| < ∈
∴ |2x – 4| < ∈
∴ |2(x – 2)| < ∈
∴ 2|x – 2| < ∈
∴ `|x - 2| < ∈/2`
∴ if we take δ = `∈/2`, then
0 < |x – 2| < δ ⇒ |f(x) – 7| < ∈
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