For the given continuous function, value of c, and ε, what is f(c) and an appropriate δ that will guarantee the continuity of f?
5x – 2 , c = 3 , and ε = 1.
Since c = 3,
f(c) = f(3) = 5(3) – 2 = 13.
Now we turn to the inequality f(c) – ε < f(x) < f(c) + ε, which for this function is, and value of ε is
13 – 1 < 5x – 2 < 13 + 1
and solve for x:
12 + 2 < 5x < 14 + 2, so
2.8 < x < 3.2
Now we can finish it up by subtracting c from all parts of the inequality to find δ.
2.8 – 3 < x – 3 < 3.2 – 3
-0.2 < x – 3 < 0.2
so δ = 0.2.
For the given continuous function, value of c, and ε, find f(c) and a value of δ such that if |x – c| < δ, |f(x) – f(c)| < ε.
ex , c = 0 , and ε = 1.
We start with the inequality f(c) – ε < f(x) < f(c) + ε and fill in what we're given for c, f, and ε:
e0 – 1 < ex < e0 + 1,
0 < ex < 2.
Now use the natural log to solve for x:
ln(0) < ln(ex) < ln(2)
ln(0) < x < ln(2)
Here we have a bit of a problem, though. We can't find ln(0); it doesn't exist. What we can say though, is that since ln(x) has approaches -∞ as x approaches 0,
-∞ < x < ln(2).
Next is where we would usually subtract c from the entire inequality. Since c = 0, that's not going to change the inequality, though, and we can just say δ is ln(2).
For the given continuous function, value of c, and ε, what is f(c) and a value of δ guaranteed by the continuity of f?
x2 + 2 , c = -2 , and ε = 0.2.
Squares have both positive and negative square roots.
We're getting the hang of this now. We always start with the inequality
f(c) – ε < f(x) < f(c) + ε
and fill in what we were given for c, f, and ε:
6 – 0.2 < f(x) < 6 + 0.2, so
5.8 < x2 + 2 < 6.2.
Solve the inequality for x. Subtract 2 from all parts of the inequality to find
3.8 < x2 < 4.2, then take square roots and round to find 1.949 < x < 2.049.
Notice this doesn't make sense. x is supposed to be close to -2, not close to 2. The problem is that we need to take negative square roots, which gives us
-1.949 > x > -2.049
(also notice that the direction of the inequalities need to switch, since -1.949 is bigger than -2.049).
Finally, subtract c from all parts of the inequality and we'll have δ. In this case c = -2, so
-1.949 – (-2) > x – (-2) > -2.049 – (-2)
0.051 > x - (-2) > -.049.
Therefore x needs to be within 0.049 of -2, so
δ = 0.049.
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