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3.3: Limits

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The notion of a limit is closely related to that of a derivative, but it is more general. In this chapter f will always be a real function of one variable. Let us recall the definition of the slope off at a point a:

S is the slope of f at a if whenever dx is infinitely close to but not equal to zero, the quotient f(a+Δx)f(a)Δxis infinitely close to S.

We now define the limit. c and L are real numbers.

Definition:

L is the limit of f(x) as x approaches c if whenever x is infinitely close to but not equal to c, f(x) is infinitely close to L.

In symbols, limxcf(x)=Lif whenever xc but xc, f(x)L. When there is no number L satisfying the above definition, we say that the limit of f(x) as x approaches c does not exist.

Notice that the limit limxcf(x)=Ldepends only on the values of f(x) for x infinitely close but not equal to c. The value f(c) itself has no influence at all on the limit. In fact, it very often happens that limxcf(x)=Lexists but f(c) is undefined.

Figure 3.3.1(a) shows a typical limit. Looking at the point (c,L) through an infinitesimal microscope, we can see the entire portion of the curve with xc because f(x) will be infinitely close to L and hence within the field of vision of the microscope.

In Figure 3.3.1(b), part of the curve with xc is outside the field of vision of the microscope, and the limit does not exist.

Our first example of a limit is the slope of a function.

Points on a curve f(x) where the limit L exists as x approaches the point c, and where the limit does not exist as x approaches c.
Figure 3.3.1: Points on a curve where limits exist and do not exist.
Theorem 3.3.1

The slope of f at a is given by the limit f(a)=limΔx0f(a+Δx)f(a)Δx.

Verbally, the slope of f at a is the limit of the ratio of the change in f(x) to the change in x as the change in x approaches zero. The theorem is seen by simply
comparing the definitions of limit and slope. The slope exists exactly when the limit exists; and when they do exist they are equal. Notice that the ratio f(a+Δx)f(a)Δx is undefined when Δx=0.

The slope of f at a is also equal to the limit f(a)=limxaf(x)f(a)xa.

This is seen by setting Δx=xa,x=a+Δx.

Then when xa but xa, we have Δx0 but Δx0; and f(x)f(a)xa=f(a+Δx)f(a)Δxf(a).

Sometimes a limit can be evaluated by recognizing it as a derivative and using Theorem 3.3.1 above.

Example 3.3.1

Evaluate limΔx0(3+Δx)29Δx.

Solution

Let F(x)=x2. The given limit is just F(3), F(3)=limΔx0F(3+Δx)29Δx=6.F(3)=23=6.

Therefore limΔx0(3+Δx)29Δx=6.

The symbol x in limxcf(x)is an example of a "dummy variable." The value of the limit does not depend on x at all. However, it does depend on c. If we replace c by a variable u, we obtain a new function L(u)=limxuf(x).

A limit limxcf(x) is usually computed as follows.

Step 1: Let x be infinitely close but not equal to c, and simplify f(x).
Step 2: Compute the standard part st(f(x)).

Conclusion: If the limit limxcf(x) exists, it must equal st(f(x)).

Example 3.3.1 (continued)

Instead of using the derivative, we can directly compute limΔx0(3+Δx)29Δx.

Solution

Step 1
Let Δx0, but Δx0. Then (3+Δx)29Δx=(9+6Δx+Δx29Δx=6Δx+Δx2Δx=6+Δx.

Step 2
Taking standard parts, st(3+Δx)29Δx=st(6+Δx)=6.

Therefore the limit is equal to 6. (See Figure \(\PageIndex{2}.)


Graph of f(Delta x) vs. Delta x, where the function f is given as the the difference of the square of 3 plus Delta x, and 9, the whole divided by Delta x. The graph approaches the limit of 6 at a Delta x value of 0.
Figure 3.3.2: Graph of f(Δx)=(3+Δx)29Δx, showing the limit is 6 as Δx approaches 0.
Example 3.3.2

Find limt4(t2+3t5).

Solution

Step 1
Let t be infinitely close to but not equal to 4.

Step 2
We take the standard part st(t2+3t5)=42+345=23, so the limit is 23. 

Example 3.3.3

Find limx2=x2+3x10x24.

Solution

Step 1 
This time the term inside the limit is undefined at x=2. Taking x2 but x2, we have x2+3x10x24=(x2)(x+5)(x2)(x+2)=x+5x+2.

Step 2
st(x2+3x10x24)=st(x+5x+2)=2+52+2=74.

Thus limx2(x2+3x10x24)=74.

Example 3.3.4

Find limx0((2/x)+3(3/x)1).

Solution

Step 1
Taking x0 but x0, (2/x)+3(3/x)1=2+3x3x.

Step 2
st((2/x)+3(3/x)1)=st(2+3x3x)=23.

Thus the limit exists and equals 23.

Example 3.3.5

Find limx9(x3x9).

Solution

Step 1
Taking x9 but x9, x3x9=(x3)(x+3)(x9)(x+3)=x9(x9)(x+3)=1x+3.

Step 2
st(x3x9)=st(1x+3)=19+3=16,so the limit exists and is 16.

Our rules for standard parts in Chapter 1 lead at once to rules for limits. We list these rules in Table 3.3.1. The limit rules can be applied whenever the two limits limxcf(x) and limxcg(x) exist.

Table 3.3.1. Rules for standard parts as applied to limits.
Standard Part Rule Limit Rule
st(kb)=k st(b), k real limxckf(x)=klimxcf(x)
st(a+b)=st(a)+st(b) limxc(f(x)+g(x))=limxcf(x)+limxcg(x)
st(ab)=st(a)st(b) limxc(f(x)g(x))=limxcf(x)limxcg(x)
st(a/b)=st(a)/st(b) limxc(f(x)/g(x))=limxcf(x)/limxcg(x), if limxcg(x)0
st(na)=nst(a), if a>0 limxcnf(x)=nlimxcf(x), if limxcf(x)>0
Example 3.3.6

Find limx1(x22x)(x21)/(x1).

Solution

All the limits involved exist, so we can use the limit rules to compute the limit as follows. First we find the limit of the expression inside the radical. 

limx1x21x1=limx1(x1)(x+1)x1=limx1(x+1)=2.

Now we find the answer to the original problem.

limx1(x22x)(x21)/(x1)=limx1(x22x)limx1(x21)/(x1)=(12)2=2.

There are three ways in which a limit limxcf(x) can fail to exist:

  1. f(x) is undefined for some x which is infinitely close but not equal to c.
  2. f(x) is infinite for some x which is infinitely close but not equal to c.
  3. The standard part of f(x) is different for different numbers x which are infinitely close but not equal to c.
Example 3.3.7

limx0x does not exist because x is undefined for negative infinitesimal x0. (See Figure 3.3.3(a).)

Example 3.3.8

limx01/x2 does not exist because 1/x2 is infinite for infinitesimal x0. (See Figure 3.3.3(b).)

Graphs of the square root function, the inverse square function, and the quotient of x and its absolute value.
Figure 3.3.3: Graphs of (a) y=x, (b) y=1x2, and (c) y=x|x|.
Example 3.3.9

limx0x/|x| does not exist because st(x|x|)={1if x>0,1if x<0.

(See Figure 3.3.3(c)).

In the above examples the function behaves differently on one side of the point 0 than it does on the other side. For such functions, one-sided limits are useful.

We say that limxc+f(x)=Lif whenever x>c and xc, f(x)L.

limxcf(x)=Lmeans that whenever x<c and xc, f(x)L. These two kinds of limits, shown in Figure 3.3.4, are called the limit from the right and the limit from the left.

A function f(x) has a discontinuity at x = c. The region of f(x) for x values less than or equal to c ends at a value of L where x = c, so the limit from the left of c is L. The region of f(x) for x values greater than or equal to c ends at a value of M where x = c, so the limit from the right of c is L.
Figure 3.3.4: One-sided limits.
Theorem 3.3.2

A limit has value L, limxcf(c)=L,

if and only if both one-sided limits exist and are equal to Llimxcf(c)=limxc+f(c)=L.

Proof

If \(\displaystyle \lim_{x \rightarrow c\) f(x) = L\), it follows at once from the definition that both one-sided limits are L.

Assume that both one-sided limits are equal to L. Let xc, but xc. Then either x<c or x>c. If x<c, then because limxcf(x)=L, we have f(x)L. On the other hand if x>c, then limxc+f(x)=L gives f(x)L. Thus in either case f(x)L. This shows that limxcf(x)=L.

When a limit does not exist, it is possible that neither one-sided limit exists, that just one of them exists, or that both one-sided limits exist but have different values.

Example 3.3.7 (continued)

limx0+x=0, and limx0x does not exist.

Example 3.3.8 (continued)

Neither limx0+1/x2 nor limx01/x2 exists.

Example 3.3.9 (continued)

limx0+x/|x|=1, and limx0x/|x|=1

Problems for Section 3.3

In each problem below, determine whether or not the limit exists. When the limit exists, find its value. With a calculator, compute some values as x approaches its limit, and see what happens.

1. limt43t2+t+1 2. limΔx1(Δx)2+2Δx+1Δx+1
3. limxccx 4. limy01y5
5. limx2xx24 6. limx2x24x2
7. limv88vv8 8. limx0x+11x
9. limu13u1u1 10. limt0t32t2+43t25t+7
11. limy0(1+1/y1/y) 12. limx0(a+x)2a2x
13. limy1y2+1y+1 14. limx1|x1|x1
15. limx1+|x1|x1 16. limxccx
17. limz1z+z+z 18. limxa|ax|
19. limx0+x1+x2 20. limx0x1+x2
21. limt01+2t134t1 22. limx03+4x15x26x1+3x2
23. limΔx0(x+Δx)2x2Δx 24. limΔx01x+Δx1xΔx (x0)
25. limΔt0t+ΔttΔt (t>0) 26. limΔt0(t+Δt)1/5t1/5Δt (t>0)
27. limΔx0(xΔx)3x3Δx 28. limΔx0x+Δxx+Δx+1xx+1Δx
29. limΔx0|(1+Δx)3(1+Δx)|Δx 30. limΔx0+|(1+Δx)3(1+Δx)|Δx
31. limΔx01(1+Δx)2Δx    

This page titled 3.3: Limits is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by H. Jerome Keisler.

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