- #1
adelin
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- 0
In summary, the conversation discusses a proof that involves a quadratic function and the use of the inequality |x + 5||x - 4| < 10|x - 4| to simplify the problem. The main point of confusion is understanding the assumptions made in the proof and how they lead to the desired conclusion.
- #2
- 22,183
- 3,324
You have a previous thread asking the same thing. Was that thread helpful? Did you understand everything there?
If so, can you start by explaining what you think they're doing? And can you explain what you don't get??
If so, can you start by explaining what you think they're doing? And can you explain what you don't get??
- #3
adelin
- 32
- 0
micromass said:
This is another proof
- #4
- 22,183
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adelin said:
It is very similar. So please, tell us what you think first.
- #5
adelin
- 32
- 0
micromass said:
in the next step they arrive to the same conclusion. If δ <1 then δ<10. ( I may be wrong)
The next step is what become problematic for me to understand.
- #6
Mark44
Mentor
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This problem is qualitatively different from the other one. In the earlier problem, the function was linear. Here the function is a quadratic.
In the second line, which is what I believe you're asking about, they make the assumption that ##\delta < 1##. Then if ##|x - 4 | < \delta < 1##, they can say that x will be between 3 and 5. Note that I'm ignoring the part where it says 0 < |x - 4|. All this does is eliminate the possibility of x being equal to 4.
Since 3 < x < 5, the largest that |x + 5| can be is 10. From this, they can write
## |x + 5||x - 4| < 10|x - 4|##
If we take ##\delta = \epsilon/10##, then when ##|x - 4| < \delta##, it will follow that
##|x + 5||x - 4| < 10|x - 4| < 10 * \delta < 10 * \epsilon/10 = \epsilon ##
In the second line, which is what I believe you're asking about, they make the assumption that ##\delta < 1##. Then if ##|x - 4 | < \delta < 1##, they can say that x will be between 3 and 5. Note that I'm ignoring the part where it says 0 < |x - 4|. All this does is eliminate the possibility of x being equal to 4.
Since 3 < x < 5, the largest that |x + 5| can be is 10. From this, they can write
## |x + 5||x - 4| < 10|x - 4|##
If we take ##\delta = \epsilon/10##, then when ##|x - 4| < \delta##, it will follow that
##|x + 5||x - 4| < 10|x - 4| < 10 * \delta < 10 * \epsilon/10 = \epsilon ##
FAQ: What is being Done in This proof of Limits?
What is being done in this proof of Limits?
In this proof of limits, we are trying to show that a function approaches a certain value as its input approaches a certain point.
What is the significance of proving limits?
Proving limits is important in mathematics and science as it helps us understand the behavior of functions and the values they approach. It also plays a crucial role in calculus and other higher level mathematical concepts.
What are the key steps in proving limits?
The key steps in proving limits include understanding the definition of a limit, determining the value that the function approaches, and using mathematical techniques such as the epsilon-delta proof to show that the limit actually exists.
How do you determine if a limit exists?
A limit exists if the left-hand limit (approaching from the left side of the input) and the right-hand limit (approaching from the right side of the input) are equal. Additionally, the function must approach the same value from both sides in order for the limit to exist.
What are some common mistakes when proving limits?
Some common mistakes when proving limits include not fully understanding the definition of a limit, assuming that a limit exists without proper proof, and not considering the behavior of the function as it approaches the input value. It is also important to be careful with algebraic manipulations and not to make any assumptions about the function.