Find d for |f(x) - (1/5)| < .05 if |x-5| < d

[Soaring Crane]

A positive number E and the limit L of a function f at a are given. Find a number d such that |f(x) - L| < E if 0 < |x-a| < d.

Lim x --> 5 (1/x) = 1/5; E = 0.05

The answer or value of d = 1/505.

These are the steps that I did, and, unfortunately, my efforts did not end in that result.

|f(x) - (1/5)| < .05 if |x-5| < d.

a. f(5 - d_1) = (1/5) + .05 = 0.25

1/(5 - d_1) = 0.25

d_1 = 1

b. f(5 + d_2) = (1/5) - 0.05 = .15

1/(5 + d_2) = 0.15

d_2 = 1.67

Neither delta fits the answer. Why is my proof terribly incorrect? Is there an alternate method?

Thanks.

 

[Nate810]

Your proof is not incorrect, but you are using two different delta values. For |f(x) - L| < E to be true, 0 < |x-a| < d must be true for the same value of d. You need to set a single value for d that satisfies both of these conditions. To do this, start by setting d to the smaller of the two deltas (in this case, d = 1). Then, check that |f(5+d) - L| < E is also true. If it is not, then increase the value of d until it is. In this example, we have |f(5+1) - L| = |(1/6) - (1/5)| = 0.16, which is not less than 0.05, so we need to increase the value of d. We can do this by setting d = 1.67 (the larger of the two deltas you calculated), which gives us |f(5+1.67) - L| = |(1/6.67) - (1/5)| = 0.04, which is less than 0.05. Therefore, the answer is d = 1.67.

 

[quicknote]

First of all, it is important to note that the value of d in this problem does not necessarily have to be a whole number. So, your calculations for d_1 and d_2 may not necessarily be incorrect.

To find the value of d, we need to use the definition of a limit. We know that the limit of f(x) as x approaches 5 is 1/5, and we want to find a value of d such that when x gets closer and closer to 5 (but not equal to 5), the difference between f(x) and 1/5 is less than 0.05.

So, we can set up the inequality:

|f(x) - 1/5| < 0.05

And this holds true if:

-0.05 < f(x) - 1/5 < 0.05

-0.05 + 1/5 < f(x) < 0.05 + 1/5

0.15 < f(x) < 0.25

Now, we can use the definition of a limit to find a value of d that satisfies this inequality. According to the definition of a limit, there exists a positive number d such that if 0 < |x - 5| < d, then |f(x) - 1/5| < 0.05.

This means that f(x) must be between 0.15 and 0.25 when x is within d units of 5. So, we can choose any value of d that satisfies this condition. For example, we can choose d = 0.01, which means that when x is between 4.99 and 5.01, f(x) will be between 0.15 and 0.25, satisfying the inequality.

In summary, the value of d is not a specific number, but rather any positive number that satisfies the condition that f(x) must be between 0.15 and 0.25 when x is within d units of 5. I hope this helps clarify the problem and your approach.