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The Formal Version Examples

Example 1

Let f(x) = 2x, c = 4, and ε = 0.5. Find an appropriate δ > 0, so that if x is within δ of 4, f(x) will be within ε of f(c).

Example 2

Let f(x) = x2, c = 2, and ε = 0.1. What is an appropriate value of δ > 0, that will guarantee that if |xc| < δ, |f(x) – f(c)| < ε?

Example 3

Let f(x) = 4x + 1. 

Find a value of δ such that if |x – 5| < δ, then |f(x) – f(5)| < 0.1.

Example 4

Let f(x) = sin(x

Find a value of δ such that if , then .

Example 5

Let . Find a value of δ such that if |x – 2| < δ, then |f(x) – 0.5| < 0.005.

Example 6

Let . What is a value of δ that will guarantee that if |x – 7| < δ, then ?

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