Limits/continuity - finding largest delta

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In summary, the question is asking to find the largest value of δ such that the inequality |x - 5| < δ => |1/x - 1/5| < 1/100 is satisfied. The person attempted to solve it by finding an upper bound for 1/x, which led to an answer of 5/21. However, they are unsure if this is correct and would appreciate any help or corrections.
  • #1
roman93
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Hey guys, I found this question and it started bugging me:

find the _largest_ δ such that |x - 5| < δ => |1/x - 1/5| < 1/100.

This is what I did to try solve the question:

From |1/x - 1/5| < 1/100 : I got 1/x > 19/100 and so I wanted x < 100/19
plugging that back into the first inequality it gave δ = 5/19

But I don't think what I did is right because when I was trying values for x which satisfied this it gave answers which were over 1/100.

So my question is how would I solve this type question.
Any help is very much appreciated :)

Thank you in advance!
 
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  • #2
I think I know where I went wrong, what I should be doing is finding an upper bound for 1/x rather than a lower bound. The answer I now have is 5/21 and this seems to work.

If I have gone wrong somewhere, please do point it out =]
 

FAQ: Limits/continuity - finding largest delta

What is the purpose of finding the largest delta in limits and continuity?

The largest delta in limits and continuity is used to determine the behavior of a function at a specific point. It helps us understand the relationship between the input and output of a function and whether it is continuous or not.

How is the largest delta calculated in limits and continuity?

The largest delta is calculated by evaluating the function at a specific point and finding the difference between the output values as the input approaches the given point. This difference, or delta, is then compared to other values of delta to determine the largest one.

What is the significance of finding the largest delta in limits and continuity?

By finding the largest delta, we can determine the maximum distance between the input and output values of a function at a specific point. This allows us to determine whether the function is continuous at that point or if there is a discontinuity present.

Can the largest delta be negative in limits and continuity?

No, the largest delta cannot be negative in limits and continuity. Delta represents the difference between two values and cannot be negative. If the largest delta is negative, it means that the input is decreasing as the output approaches the given point, which is not possible in a continuous function.

How does finding the largest delta help in determining the continuity of a function?

By comparing the largest delta to other values of delta, we can determine the smoothness of a function at a specific point. If the largest delta is equal to zero, it means the function is continuous at that point. However, if the largest delta is greater than zero, it indicates a jump or a break in the function and therefore, a discontinuity.

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