Math 1271, Dis 042/046, Examples for 2.4:

Example 1

Using the graph of f(x) = x^(1/3) below, find a delta such that:
      if   |x - 8| < delta,    then   |f(x) - 2 | < .5

graph of x^(1/3)

Solution to Example 1

We need to pick a delta so that if x is in a delta-neighborhood of 8, f(x) is between 1.5 and 2.5. We look at the handy lines already drawn on the graph, and see that delta = 4 will work. (Other deltas will work too, but delta can't be bigger than the distance between 8 and 3.375, which is where the leftmost vertical line hits the x-axis.)

Example 2

Using the graph of f(x) = x^2 below, find a delta such that:
      if   |x - 2| < delta,    then   |f(x) - 4 | < .5

graph of x^(1/3)

Solution to Example 2

We need to pick a delta so that if x is in a delta-neighborhood of 2, f(x) is between 3.5 and 4.5. We look at the handy lines already drawn on the graph, and see that delta = .1 will work. (Again, other deltas will work too.)

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