Precise Definition of a Limit, Example Clarification

In summary, the conversation revolves around a student having difficulty understanding an example in their textbook. The example involves finding a specific value when given a range for x. The student is confused about the use of numbers in the solution and how they were obtained. However, the main takeaway is that if x is within a certain range, the corresponding value for f(x) can be determined.
  • #1
Chase.
12
0
This isn't a homework problem. My textbook has an example for this subject and I'm having difficulty understanding it.

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I follow the example up until the point at which it says, "Notice that 0 < | x - 3 | < (0.1)/2 = 0.05, then "

I don't understand why delta was substituted with (what seem to be) arbitrary numbers.
 
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  • #2
It looks like the 0.05 came out of thin air because they didn't show you how they found it. You'll probably have practice doing that soon in your homework.

The point was, however, that if x is within 0.05 of 3, then f(x) will be within 0.1 of 5, which answers the question posed earlier.
 

FAQ: Precise Definition of a Limit, Example Clarification

1. What is the precise definition of a limit?

The precise definition of a limit in calculus is as follows: Given a function f(x), a point c, and a real number L, the limit of f(x) as x approaches c is L if for every positive number ε, there exists a positive number δ such that if 0 < |x-c| < δ, then |f(x)-L| < ε.

2. What is the significance of the precise definition of a limit?

The precise definition of a limit allows us to rigorously determine the behavior of a function f(x) as x approaches a specific point c. It also allows us to prove the existence or non-existence of a limit, and to accurately evaluate limits using mathematical techniques.

3. Can you provide an example to clarify the precise definition of a limit?

Consider the function f(x) = x^2 and the point c = 2. To prove that lim x->2 (x^2) = 4, we must show that for any given ε > 0, we can find a corresponding δ such that if 0 < |x-2| < δ, then |x^2 - 4| < ε. In this case, we can choose δ = min{1, ε/5} to satisfy the definition.

4. What happens if the limit does not exist?

If the limit does not exist, it means that the function does not approach a single value as x gets closer and closer to the given point. This could be due to a variety of reasons, such as a jump or discontinuity in the function, or the function approaching different values from different directions.

5. How is the precise definition of a limit different from the intuitive concept of a limit?

The intuitive concept of a limit refers to the idea that as x gets closer and closer to a given point, c, the values of f(x) get closer and closer to a specific value, L. The precise definition adds a level of mathematical rigor by specifying that for any given margin of error, ε, there exists a corresponding margin of input, δ, that will result in values of f(x) within ε of L.

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