Indeterminate Case

We will see that this same pattern occurs in many of the limit problems that we will tackle. In the last lecture we saw how we may calculate f c as one step towards trying to determine what the limit of f x is as x approaches c. In cases where f c exists, this is simple, because then the limit will be equal to f c. However, in most cases we are calculating the limit precisely because f c does NOT exist, and in those cases, calculating f c will always produce an undefined expression.

However, neither the top nor the bottom of the fraction ever actually reach zero. In other words, both the top and the bottom of f x are shrinking in magnitude as x gets closer and closer to c. However, neither the first nor the second term of the expression ever actually reach infinity, because that is not possible. Infinity is not a number that can be reached. In other words, both the first and second term in f x are growing unbounded in magnitude as x gets more and more negative. In other words, as x approaches -2 from the right, the numerator of f x becomes very close to -2 while the denominator's magnitude grows ever smaller.

If we divide numbers that are arbitrarily close to -2 by positive numbers with increasingly smaller magnitude, we get negative numbers with increasingly larger magnitude as a result. And if put numbers in the denominator with a small enough magnitude, we can get numbers with as large a magnitude as we want - so this behavior is unbounded. This limit will turn out to be a specific finite number that is not zero in this case. In other words, as x approaches 0, the magnitudes of both the numerator and the denominator of f x grow ever smaller. This is not enough information for us to conclude anything about the limit, because dividing numbers with increasingly smaller magnitude by other numbers with increasingly smaller magnitude can produce a number of different results: it depends on how "small" the magnitude of the numerator is in comparison to the denominator!

And we don't know anything about the relationship between the numerator and the denominator yet. In this case, because f x contains a radical in the numerator, one possible approach is for us to try to rewrite the expression so that the radical in the numerator is canceled out - this may then allow us to cancel something out of the top and bottom of the fraction.

Big Idea: Algebra is just a way of Looking for Structure. In this case, two equations are equivalent if they have the same solution set the same values of x that make the equation true. Right now we are interested in finding limits, and the only way we know how to find limits so far is to simply plug in c for x , and to calculate f c. But sometimes this doesn't work - sometimes just plugging in c for x gives us something that is not defined, like.

L'Hôpital's rule example 1 - Derivative applications - Differential Calculus - Khan Academy

This limit will turn out to be zero in this case. Again, as x approaches 0, the magnitudes of both the numerator and the denominator of f x grow ever smaller, and this is not enough information for us to conclude anything about the limit, because it depends on how "small" the magnitude of the numerator is in comparison to the denominator , and therefore depends on the relationship between the numerator and the denominator.

In this case, because f x contains a radical in the denominator, one possible approach is for us to try to rewrite the expression so that the radical in the denominator is canceled out - this may then allow us to cancel something out of the top and bottom of the fraction. In this case the limit will not exist because f x will decrease without bound as x approaches 0 from the left, and increase without bound as x approaches 0 from the right.

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Small eigenvalues of large Hankel matrices: The indeterminate case

This problem is similar to example 1 above. In other words, as x approaches 0 from the left, the numerator of f x becomes very close to 1 while the denominator's magnitude grows ever smaller, and dividing a relatively fixed positive number like 1 by negative numbers with increasingly smaller magnitude, we get negative numbers with increasingly larger magnitude as a result. And as in example 1, this behavior is unbounded because by making the magnitude of the denominator small enough, we can get numbers with as big a magnitude as we want.

Let's think back about these four examples, and summarize the differences between these four somewhat similar problems. In each of these examples f x was a fraction that had a zero in the denominator when we substituted c in for x , but in each case, the numerator and denominator of f x had different relationships as x got closer and closer to c :. In example 1, as x got ever closer to c , the numerator of f x approached a fixed number, while the magnitude of the denominator shrank indefinitely.

This resulted in numbers with a magnitude that increased without bound because dividing a relatively fixed value with numbers that are ever closer to zero results in numbers with an increasingly large magnitude, and because by making the magnitude of the denominator small enough, we can get numbers with as big a magnitude as we want. In example 2, 3, and 4, as x got ever closer to c , the magnitudes of both the numerator and the denominator of f x shrank indefinitely. In this case, it is not really possible for us to see that they will "shrink at roughly the same rate" directly; we are only able to determine this by first rewriting the function using algebra.

In example 3, the magnitude of the numerator shrank much more quickly than the magnitude of the denominator, so that when dividing the numerator by the denominator we get values that are ever-closer to zero. In this case, it can be difficult to see that the magnitude of the numerator will shrink "much more quickly", but again, we are able to determine that this is the case by rewriting the function using algebra.

In example 4, the magnitude of the denominator shrank much more quickly than the magnitude of the numerator, so that when dividing the numerator by the denominator we get values that have an ever-larger unbounded magnitude. In this case, it can be difficult to see that the magnitude of the denominator will shrink "much more quickly", but again, we are able to determine that this is the case by rewriting the function using algebra.

This leads us to a few definitions that we will use to describe this distinction:. Notice that these definitions only have meaning when we are calculating a limit. It would be incorrect in that context for me to say anything about determinate or indeterminate forms, because I am not calculating a limit! Now let's return to some more examples that give us other undefined expressions when we calculate f c , and let's see if we can determine which undefined values for f c are indeterminate versus determinate forms! In other words, as x decreases without bound i. If we divide numbers that are arbitrarily close to 3 by negative numbers with increasingly larger magnitude, we get negative numbers with increasingly smaller magnitude as a result.

So we get numbers that are closer and closer to zero. In other words, as x increases without bound, the denominator of f x becomes very close to 0 while the numerator's magnitude grows ever larger. If we divide positive numbers that have an increasingly larger magnitude by positive numbers with an increasingly smaller magnitude i.

In other words, as the magnitude of x grows without bound, the magnitudes of both the numerator and the denominator of f x grow ever larger.

Mathematics > Classical Analysis and ODEs

This is not enough information for us to conclude anything about the limit, because dividing numbers with increasingly larger magnitude by other numbers with increasingly larger magnitude can produce a number of different results: it depends on how "large" the magnitude of the numerator is in comparison to the denominator! For this, we ask ourselves if there is any algebra that we could use to rewrite f x without changing its value for negative x -values with particularly large magnitude.

So we need to think of something that we can do to "rewrite" f x so that we can get rid of the x 's in either the numerator or the denominator. There is also more than one way to solve this problem. The method used above is just one example, but the limit can also be found in a different way, using a similar algebraic technique, but this time dividing by the largest power of x overall, instead of just the largest power of x in the numerator.

Notice that either method works equally well in helping us to find the limit, giving us the same answers in both cases:. This limit will turn out to be 2.

Just as before, as the magnitude of x grows without bound, the magnitudes of both the numerator and the denominator of f x grow ever larger, and once again this is not enough information for us to conclude anything about the limit, because we don't know anything about the relationship between the numerator and the denominator yet. But these aren't the only two examples of formats that produce determinant and indeterminate forms.

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There are a number of other determinant and indeterminate forms which we will encounter when working on trying to solve limit problems algebraically. Here is a table which shows all of the determinant and indeterminate forms:. So any time we calculate f c by plugging in c for x , when our goal is really to find the limit of f x as x approaches c, we know that if the result is on the list of indeterminate forms above, we will need to do more work before we can calculate the limit usually by rearranging f x using some algebra. An explicit lower bound is given in terms of the orthonormal polynomials and we find expressions for this lower bound in a number of indeterminate moment problems.

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limits - Is $\sqrt{-\infty}$ one of the indeterminate cases? - Mathematics Stack Exchange

Save to Library. Create Alert. Figures and Topics from this paper. Citations Publications citing this paper. A reproducing kernel condition for indeterminacy in the multidimensional moment problem Roger A. Recent developments on the moment problem Gwo Dong Ming Lin. Weems , Yang Chen. References Publications referenced by this paper. Ismail , David Masson.

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