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Section 5.4 Indeterminate Form và L"Hôpital"s Rule
¶Subsection 5.4.1 Indeterminate Forms
¶Before we embark on introducing one more limit rule, we need lớn recall a concept from algebra. In your work with functions (see Chapter 2) and limits (see Chapter 4) we sometimes encountered expressions that were undefined, because they either lead khổng lồ a contradiction or to lớn numbers that are not in the phối of numbers we started out with. Let us look at an example for either scenario khổng lồ investigate the concept “undefined” more deeply.
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Suppose that
eginequation*f(x)=frac1x ext.endequation*
What happens when (x=0 ext?) Then (f(0)=1/0 ext,) but (1/0) is undefined. Why is that? Let"s assume this value is defined. This means that (1/0) is equal to lớn some number, hotline it (n ext.) Then
eginequation*eginsplit frac10 amp = n \ 1 div 0 amp = n \ 1 amp = n imes 0 \ 1 amp = 0 endsplitendequation*
Clearly, 1 is not equal lớn 0, and so this statement is a contradiction. In fact, if we analyze the satament
eginequation*1 = n imes 0 ext,endequation*
we notice that there is no number for (n) that will satisfy this equation. Therefore, (1/0) could not have been a number, and hence we say (1/0) is undefined. This is the reason why we write that the tên miền of (f) is given by
eginequation*mathcalD_f =left x in mathbbRig vert x eq 0 ight\text.endequation*
Example 5.31. Different Number Set.
Suppose that (f(x)=sqrtx-1) và that we are working over the real numbers. What happens when (x=0 ext?) Then
eginequation*f(0) = sqrt-1 ext,endequation*
but (sqrt-1) is undefined over the real numbers. Why is that? Let"s assume this value is defined. Then by the definition of square root, there is a real number (n) such that (-1 = n^2 ext.) Clearly, the square of a real number cannot produce a negative real number because positive × positive & negative × negative are both positive real numbers. In fact, (sqrt-1) is the imaginary number (i ext,) which belongs to lớn the phối of complex numbers.
When we work out limit problems algebraically, we will often get as an initial answer something that is undefined. This is because the places where a function is undefined are the “interesting” places to lớn look for limits. For example, if
eginequation*g(x)=fracx^2-9x-3 ext,endequation*
then
eginequation*g(3) = frac3^2-93-3 = frac00 ext,endequation*
but
eginequation*lim_x o 3 g(x) = lim_x o 3 fracx^2-9x-3 = lim_x o 3 frac(x-3)(x+3)x-3 = lim_x o 3 left(x+3 ight) = 6 ext.endequation*
The function (g) is a line with a hole at (x=3) và the limit showed us that this hole can be removed with the (y)-value 6 at (x=3) (see Fig 5.11).
Figure 5.11. The function (g(x)=fracx^2-9x-3) is undefined at (x=3 ext.)
However, we must remember that when we are calculating the limit of (f(x)) as (x o a) we are not interested in the behavior of (f(x)) at (a ext,) but we want to know the behavior of (f(x)) around (a ext.) It is therefore important for us lớn identify an undefined value a of a function, & furthermore, khổng lồ investigate whether the type of undefined value can tell us something about the behavior of the function around (a ext.)
Before we continue, we need lớn draw attention khổng lồ a notation that we have been using when calculating limits. When we write (f(x) o 0) as (x o a ext,) we actually mean that (f(x)) gets arbitrarily close lớn zero as (x) gets closer & closer to lớn (a ext.) However, the function value never reaches zero. Similarly, when we write (f(x) o infty) as (x o a ext,) we actually mean that (f(x)) grows ever larger, without bound as (x) gets closer and closer to lớn (a ext.) However, the function value never reaches infinity, since infinity is not even a number.
Limit Behaviour.When calculating limits,
0 represents a number arbitrarily close khổng lồ zero;
(+infty) represents an arbitrarily large positive number; and
(-infty) represents an arbitrarily large negative number.
Therefore, (f(x) o frac00) as (x o a) means that (f(x)) is a fraction for which both the numerator & the denominator get arbitrarily close to zero as (x) gets closer and closer khổng lồ (a ext,) và (f(x) o fracinftyinfty) as (x o a) means that (f(x)) is a fraction for which both the numerator and the denominator grow ever larger, without bound as (x) gets closer and closer to (a ext.) We also know from experience that some limits that demonstrate (frac00) or (fracinftyinfty) behaviour work out khổng lồ be real numbers, i.e. The limit exists, while others vày not, as the following four examples remind us:
Example 5.32. Limit exists when 0/0.eginequation*lim_x o 3 fracx^2-9x-3 stackrelfrac00= lim_x o 3 frac(x-3)(x+3)x-3 =lim_x o 3 (x+3) = 6endequation*
eginequation*eginsplitlim_x o 0^+ fracsqrtx+1-1x^2 stackrelfrac00= amp lim_x o 0^+ fracsqrtx+1-1x^2 cdot fracsqrtx+1+1sqrtx+1+1 \ amp= lim_x o 0^+ frac1xleft(sqrtx+1+1 ight) stackrelfrac10^+= infty endsplitendequation*
eginequation*lim_x o infty frac1-x2x stackrelfrac-inftyinfty= lim_x o infty fracfrac1x-12 = -frac12endequation*
eginequation*lim_x o infty frac1-x^22x stackrelfrac-inftyinfty= lim_x o infty fracfrac1x-x2 = -inftyendequation*
Upon closer inspection of the undefined expressions 0/0 and (infty)/(infty ext,) we should realize that both terms are based on the division operation & ask ourselves whether there are other undefined expressions that we may encounter when taking limits. We therefore investigate arithmetic ((a+b ext,)(a-b ext,)(ab ext,)(a/b)) and exponentiating ((a^b)) operations where (a) & (b) are values that approach 0, 1, some arbitrary number (n eq 0,1) or (infty ext.) We leave it up lớn the reader to lớn perform an exhaustive listing of all combinations, & instead limit ourselves lớn the combinations that are of interest as shown in Table 5.12.
eginequation*eginarraycccccc0+0 amp infty+infty amp 0 cdot infty amp n cdot infty amp 0^0 amp 0^infty \<1ex>0-0 amp infty-infty amp frac0infty amp fracninfty amp 1^0 amp 1^infty \<1ex>0 cdot 0 amp pminfty cdot pminfty amp fracinfty0 amp fracinftyn amp n^0 amp n^infty \<1ex>frac00 amp fracpminftypminfty amp amp amp infty^0 amp infty^inftyendarrayendequation*
Table 5.12. Arithmetic and Exponentiating Combinations.
We now encourage the reader lớn investigate each one of the terms shown in Table 5.12 and decide whether the undefined expression resolves khổng lồ give a single number value or infinity (determinate form), or whether this cannot be determined (indeterminate form), all the while keeping in mind our earlier discussion on limit behaviour around (x=a ext.) We formally define this new terminology before we explore some terms together.
Definition 5.36. Determinate và Indeterminate Forms.An undefined expression involving some operation between two quantities is called a determinate khung if it evaluates to lớn a single number value or infinity.
An undefined expression involving some operation between two quantities is called an indeterminate form if it does not evaluate lớn a single number value or infinity.
We will inspect multiplication more closely. Consider (0 imes 0 ext.) Clearly, a number that is getting arbitrarily close to lớn zero that is multiplied by another number that is getting arbitrarily close to lớn zero gets even closer to lớn zero, i.e. (0 imes 0 o 0 ext.) Now consider (infty imes infty ext.) Here, multiplying two values that are growing large without bound simply means that their sản phẩm grows large without bound, i.e. (infty imes infty o infty ext.) Similarly, (left(-infty ight) imesinfty) means that the magnitude of the sản phẩm grows large without bound and that (left(-infty ight) imesinfty o -infty ext.) What about (n imesinfty ext,) when (n eq 0 ext?) Here we need lớn differentiate between negative and positive values of (n ext:) If (n>0 ext,) then (n imes infty o infty ext,) and if (nlt 0 ext,) then (n imesinfty o -infty ext.) So far, we have only encountered determinate forms involving multiplication. Lastly, consider (0 imesinfty ext.) Here, we have a number that is getting arbitrarily close lớn zero being multiplied with a value that is growing large without bounds. This is lượt thích two ends of a rope being tugged and we do not know which side is going to win. Therefore, (0 imesinfty) is an expression that cannot be determined.
We leave the remaining terms up to the reader to investigate & simply present the determinate and indeterminate forms of the expressions from Table 5.12 in Table 5.13.
eginequation*eginarraycc extbf Determinate Forms amp extbf Indeterminate Forms\ \<0.25em>0+0 amp infty - infty \<1em>0-0 amp dfrac00 \<1em>0cdot 0 amp dfracpm inftypm infty \<1em>pm infty cdot pm infty amp 0 cdot infty \<1em>dfrac0infty,dfracninfty amp 0^0 \<1em>dfracinfty0,dfracinftyn amp infty^0\<1em>ncdotinfty scriptstyle n eq 0 amp 1^infty\<1em>0^infty amp \<1em>n^infty scriptstyle n eq 1 amp \<1em>infty^infty ampendarrayendequation*
Table 5.13. Determinate và Indeterminate Forms.
Subsection 5.4.2 L"Hôpital"s Rule for Finding Limits
¶We are now in a position to lớn introduce one more technique for trying lớn evaluate a limit.
Definition 5.37. Limits of the Indeterminate Forms (frac00) and (fracinftyinfty).A limit of a quotient (limlimits_x ightarrow afracfleft( x ight) gleft( x ight) ) is said to be an indeterminate size of the type(frac00) if both (fleft( x ight) ightarrow 0) and (gleft( x ight) ightarrow 0) as (x ightarrow a ext.) Likewise, it is said to lớn be an indeterminate khung of the type (fracinfty infty ) if both (fleft( x ight) ightarrow pm infty) and (gleft( x ight) ightarrow pm infty) as (x ightarrow a) (Here, the two (pm) signs are independent of each other).
Theorem 5.38. L"Hôpital"s Rule.For a limit (limlimits_x ightarrow afracfleft( x ight) gleft( x ight) ) of the indeterminate form (frac00) or (fracinfty infty ext,)
eginequation*limlimits_x ightarrow afracfleft( x ight) gleft( x ight) =limlimits_x ightarrow afracf^prime left( x ight) g^prime left( x ight) endequation*
if (limlimits_x ightarrow afracf^prime left( x ight) g^prime left( x ight) ) exists or equals (infty) or (-infty ext.)
This theorem is somewhat difficult to lớn prove, in part because it incorporates so many different possibilities, so we will not prove it here.
Note:There may be instances where we would need lớn apply L"Hôpital"s Rule multiple times, but we must confirm that (limlimits_x o adfracf"(x)g"(x)) is still indeterminate before we attempt khổng lồ apply L"Hôpital"s Rule again.
L"Hôpital"s Rule is also valid for one-sided limits & limits at infinity.
Notation when Applying L"Hôpital"s Rule.We use the symbol (Heq) lớn denote we are using l"Hôpital"s Rule in that step.
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Compute (dslim_x o pifracx^2-pi^2sin x ext.)
eginequation*lim_x o pifracx^2-pi^2sin xHeq lim_x o pifrac2xcos x ext,endequation*
provided the latter exists. But in fact this is an easy limit, since the denominator now approaches (-1 ext,) so
eginequation*lim_x o pifracx^2-pi^2sin x=frac2pi-1 = -2pi ext.endequation*