lim
x---> 2 4x+1/3x-4
illustrate definition 2 by finding values of delta that correspond to epsilon=0.5 and epsilon= 0.1
I having trouble setting this up.
do i set up two different functions then divide?
0.2898 that's the answer i got....
To illustrate definition 2 of the limit, you need to find values of delta (denoted as δ) that correspond to given values of epsilon (denoted as ε). In this particular case, you are given two different values of epsilon: ε = 0.5 and ε = 0.1.
Definition 2 of the limit states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |(4x + 1)/(3x - 4)| < ε.
To find the values of delta corresponding to these epsilons, you will set up two separate inequalities and solve for delta in each case.
For ε = 0.5:
Step 1: Start with the given inequality: |(4x + 1)/(3x - 4)| < 0.5
Step 2: Manipulate the inequality to isolate |x - 2|:
0 < |(4x + 1)/(3x - 4)| - 0.5
Step 3: Simplify and solve for |x - 2|:
0 < |(4x + 1) - 0.5*(3x - 4)|
0 < |8x + 2 - 1.5x + 2| (distributing -0.5)
0 < |6.5x + 4|
Now, you need to find the delta value. To do this, you need to bound |6.5x + 4| by 0.5.
Step 4: Set up the inequality:
|6.5x + 4| < 0.5
Step 5: Solve the inequality for x:
-0.5 < 6.5x + 4 < 0.5
Step 6: Isolate x:
-4.5 < 6.5x < -3.5
Step 7: Divide by 6.5:
-0.692 < x < -0.538
Therefore, for ε = 0.5, the values of delta that satisfy the inequality are -0.692 < x - 2 < -0.538, which means the values of delta lie within the interval (-0.692 + 2, -0.538 + 2) or (1.308, 1.462).
Now let's find the values of delta corresponding to ε = 0.1:
Step 1: Start with the given inequality: |(4x + 1)/(3x - 4)| < 0.1
Step 2: Manipulate the inequality to isolate |x - 2|:
0 < |(4x + 1)/(3x - 4)| - 0.1
Step 3: Simplify and solve for |x - 2|:
0 < |(4x + 1) - 0.1*(3x - 4)|
0 < |6x + 2 - 0.3x + 0.4| (distributing -0.1)
0 < |5.7x + 2.4|
To find delta, you need to bound |5.7x + 2.4| by 0.1.
Step 4: Set up the inequality:
|5.7x + 2.4| < 0.1
Step 5: Solve the inequality for x:
-0.1 < 5.7x + 2.4 < 0.1
Step 6: Isolate x:
-2.5 < 5.7x < -2.3
Step 7: Divide by 5.7:
-0.439 < x < -0.404
Therefore, for ε = 0.1, the values of delta that satisfy the inequality are -0.439 < x - 2 < -0.404, which means the values of delta lie within the interval (-0.439 + 2, -0.404 + 2) or (1.561, 1.596).
It seems that the value you mentioned as the answer (0.2898) doesn't match the explanation and calculations above. Please double-check your work or provide more context for clarification.