Continuity of Functions Exercises

Answer

Since we're looking at continuity near x = 5, 5 is our value of c and f(c) = f(5) = 21. Graph the function and set the window so that

0 ≤ x ≤ 10 and 0 ≤ y ≤ 42.

We do this so that (5, 21) is in the center of the window. Now start narrowing the values of x, keeping 5 in the center, until the function goes out the sides of the graph instead of the top and bottom. Here's one possible progression, where we first bring x within 1 step of 5, then within 0.5 of 5, then within 0.25 of 5, and finally within 0.125 of 5:

If we restrict x so that 4.875 ≤ x ≤ 5.125, we find what we want: a picture where (5, 21) is in the middle, and all the values of f(x) are between 20.5 and 21.5.

Answer

If we try to get f(x) within 0.5 of f(1) = 2 (that is, 1.5 < f(x) < 2.5), we run into trouble. No matter how much we restrict x, since we aren't allowed to have 1 ≤ x ≤ 1, the picture will always have some values of x that are less than 1. Since f(x) will be less than 1 for these values, f(x) will not be between 1.5 and 2.5. We just can't get the picture we want. Sniff.

Answer

1. 2.8 ≤ x ≤ 3.2 will work, as will anything more restricted. (We can require x be 0.2 away from 3, or closer, but farther away won't work.)
2. The largest range of values that will work here is ln(0.9) ≤ x < ≤ ln(1.1). We know this because e^ln(0.9) = 0.9 and e^ln(1.1) = 1.1.
3. Trial and error is best here. Tinker with some x-values until you find a range that works. One possible range of values is -2.1 ≤ x ≤ -1.9.