Let f(x)= (x^2 -4)/(x-2) and note that lim as x approaches 2 , f(x)=4. For each value of E, use a graphing utility to find all values whenever 0<|x-2|< delta . looks like an S.
a) e=2
b) e=1
c) For any e>0, make a conjecture about the value of delta S that satisfies the preceding inequality.
How would you do this through the calculator? Can you write the steps for each part a), b), and C)
I think the first one is 0<Sdelta<or =to 2
b)?
c)?
To determine the values of delta when 0 < |x - 2| < E and plot them on a graphing utility, you can follow these steps:
a) e = 2:
1. Open a graphing utility program or calculator that allows you to plot functions.
2. Enter the function f(x) = (x^2 - 4)/(x - 2) into the calculator.
3. Plot the function on the graphing utility.
4. Draw two horizontal lines at y = 2 + e and y = 2 - e on the same graph.
5. Find the x-values at which the graph of f(x) intersects these two lines.
- The point of intersection on the right side of the vertical asymptote (x = 2) will be your upper bound for delta.
- The point of intersection on the left side of the vertical asymptote will be your lower bound for delta.
b) e = 1:
1. Follow steps 1 to 3 from part a).
2. Draw two horizontal lines at y = 2 + e and y = 2 - e, where e = 1.
3. Identify the x-values at which the graph of f(x) intersects these two lines to determine the bounds for delta.
c) Conjecture:
Based on the observations from parts a) and b), you can make a conjecture about the value of delta S that satisfies the inequality 0 < |x - 2| < E for any e > 0.
- The graph of f(x) looks like an "S" shape near x = 2.
- As e decreases, the bounds for delta on either side of x = 2 become closer together.
- The conjecture is that as e approaches 0, the value of delta S also approaches 0 so that the inequality becomes 0 < |x - 2| < 0, which is not possible.