In this new expression for q, there is no longer a denominator with t> in it. Since (1 + d) -> 2 as u -» 1 from either side, we may then conclude that lim q = 2.

Example 3 Given q = (2c + 5)/(v + 1), find lim q. The variable v again

appears in both the numerator and the denominator. If we let v —► + oo in both, the result will be a ratio between two infinitely large numbers, which does not have a clear meaning. To get out of the difficulty, we try this time to transform the given ratio to a form in which the variable v will not appear in the numerator.* This, again, can be accomplished by dividing out the given ratio. Since (2v + 5) is not evenly divisible by (v + 1), however, the result will contain a remainder term as follows:

But, at any rate, this new expression for q no longer has a numerator with v in it. Noting that the remainder 3/(u + 1) -> 0 as v -» + oo, we can then conclude that lim q = 2.

There also exist several useful theorems on the evaluation of limits. These will be discussed in Sec. 6.6.

Formal View of the Limit Concept

The above discussion should have conveyed some general ideas about the concept of limit. Let us now give it a more precise definition. Since such a definition will make use of the concept of neighborhood of a point on a line (in particular, a specific number as a point on the line of real numbers), we shall first explain the latter term.

For a given number L, there can always be found a number (L — ax) < L and another number (L + a2) > L, where a, and a2 are some arbitrary positive numbers. The set of all numbers falling between (L — ax) and (L + a2) is called the interval between those two numbers. If the numbers (L — a,) and (L + a2) are included in the set, the set is a closed interval, if they are excluded, the set is an open interval. A closed interval between (L — ax) and (L + a2) is denoted by the bracketed expression

[L — at, L + a2] = {q\L — at<q<L + a2) and the corresponding open interval is denoted with parentheses: (6.4) (L - ax, L + a2) = [q \ L - a, < q < L + a2)

* Note that, unlike the t> -> 0 case, where we want to take v out of the denominator in order to avoid division by zero, the v -» oo case is better served by taking v out of the numerator. As v -> oo, an expression containing v in the numerator will become infinite but an expression with v in the denominator will, more conveniently for us, approach zero and quietly vanish from the scene.

Thus, [ ] relate to the weak inequality sign < , whereas ( ) relate to the strict inequality sign < . But in both types of intervals, the smaller number (L — ax) is always listed first. Later on, we shall also have occasion to refer to half-open and half-closed intervals such as (3,5] and [6, oo), which have the following meanings:

(3,5] = {x | 3 < x < 5} [6, oo) = {x | 6 < x < oo}

Now we may define a neighborhood of L to be an open interval as defined in (6.4), which is an interval "covering" the number L* Depending on the magnitudes of the arbitrary numbers ax and a2, it is possible to construct various neighborhoods for the given number L. Using the concept of neighborhood, the limit of a function may then be defined as follows:

As v approaches a number N, the limit of q = g(v) is the number L, if, for every neighborhood of L that can be chosen, however small, there can be found a corresponding neighborhood of N (excluding the point v = N) in the domain of the function such that, for every value of t; in that A'-neighbor-hood, its image lies in the chosen ¿-neighborhood.

This statement can be clarified with the help of Fig. 6.3, which resembles Fig. 6.2a. From what was learned about the latter figure, we know that lim q = L in

Fig. 6.3. Let us show that L does indeed fulfill the new definition of a limit. As the first step, select an arbitrary small neighborhood of L, say, (L — ax, L + a2). (This should have been made even smaller, but we are keeping it relatively large to facilitate exposition.) Now construct a neighborhood of N, say, (N - bx, N + b2), such that the two neighborhoods (when extended into quadrant I) will together define a rectangle (shaded in diagram) with two of its corners lying on the given curve. It can then be verified that, for every value of v in this neighborhood of N (not counting v = N), the corresponding value of q = g(v) lies in the chosen neighborhood of L. In fact, no matter how small an ¿-neighborhood we choose, a (correspondingly small) ¿V-neighborhood can be found with the property just cited. Thus L fulfills the definition of a limit, as was to be demonstrated.

We can also apply the above definition to the step function of Fig. 6.2 c in order to show that neither Lx nor L2 qualifies as lim q. If we choose a very small v->N

neighborhood of L,—say, just a hair's width on each side of L,—then, no matter what neighborhood we pick for N, the rectangle associated with the two neighborhoods cannot possibly enclose the lower step of the function. Consequently, for any value of v > N, the corresponding value of q (located on the lower step) will not be in the neighborhood of Lx, and thus Lx fails the test for a limit. By similar reasoning, L2 must also be dismissed as a candidate for lim q. In fact, in this case no limit exists for q as v -* N. V~'N

* The identification of an open interval as the neighborhood of a point is valid only when we are considering a point on a line (one-dimensional space). In the case of a point in a plane (two-dimensional space), its neighborhood must be thought of as an area, say, a circular area around the point.

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