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Deathfish
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In limits, how do you tell immediately if the only way to solve the problem is to test it out? What are some cases where rewrtiting the equation won't get results?
eg. 0.99, 0.999, 0.9999
eg. 0.99, 0.999, 0.9999
In limits, how do you tell immediately if the only way to solve the
- Thread starter Deathfish
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In summary, in limits, the only way to solve a problem is not by testing it out but by determining the general term and using that to prove the limit. Some cases where rewriting the equation may not yield results include when the pattern changes suddenly or when the person solving the problem is not familiar with sequences. Recognizing patterns and being able to write the general term are important skills in mathematics for finding limits.
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satyasaichand
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i didn't get you Mr. in limits we just approach a constant or a value closer than the closest possibility... this is the only concept of limits in my view...
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tiny-tim
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Hi Deathfish!
Yes, I don't get you either.
(and welcome to PF, satyasaichand!)
Yes, I don't get you either.
(and welcome to PF, satyasaichand!
)- #4
HallsofIvy
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If I am interpreting this correctly, strictly speaking you never can find a limit by "testing it out" (which I interpret to mean writing out "enough" terms to determine by inspection what it converges to). It is quite possible that the sequence is .99, .999, .9999 (1- .1^(n+1) is the general term) for the first, say, million terms then changes to some other rule entirely. You must be able to determine what the general term is.]
(Problems where you are given a finite sequence of numbers and expected to determine the general term are common in mathematics- the ability to recognise a pattern is an important skill in mathematics- but, strictly speaking, they should state that the pattern does not change.)
Assuming that the pattern in .99, .999, .9999, .99999, etc. continues then you can see that it is approaching 1 by 'looking at it', but if you want to prove that, you should rewrite it as [itex]1- .1^{n+1}[/itex]. Whether or not you can recognise a limit "by inspection" depends on how familiar you are with sequences. It is proving that limit that typically requires writing the general term in some specific form.
1. How do you define limits in mathematical terms?
Limits are defined as the value that a function approaches as the input values get closer and closer to a specific point, without actually reaching that point. This point is known as the limit point or the point of approach.
2. What is the difference between one-sided and two-sided limits?
A one-sided limit only considers the values approaching the limit point from one direction, either from the left or the right. A two-sided limit considers the values approaching from both directions and requires that the limit from both sides be equal for the overall limit to exist.
3. How do you determine if a limit exists?
A limit exists if the function approaches the same value from both sides of the limit point, or if the limit approaches infinity or negative infinity. If the function approaches different values from the left and right sides, the limit does not exist.
4. Can limits be used to find the value of a function at a specific point?
No, limits cannot be used to find the value of a function at a specific point. They only describe the behavior of the function as the input values approach a specific point, not the actual value of the function at that point.
5. What are some common techniques for solving limits?
Common techniques for solving limits include direct substitution, factoring, rationalization, and using special trigonometric identities. L'Hopital's rule and the squeeze theorem are also frequently used to solve more complex limits.
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