Graphical Illustrations

Let us illustrate, in Fig. 6.2, several possible situations regarding the limit of a function

Figure 6.2a shows a smooth curve. As the variable v tends to the value /V from either side on the horizontal axis, the variable q tends to the value L. In this case, the left-side limit is identical with the right-side limit; therefore we can write lim q = L,

Figure 6.2a shows a smooth curve. As the variable v tends to the value /V from either side on the horizontal axis, the variable q tends to the value L. In this case, the left-side limit is identical with the right-side limit; therefore we can write lim q = L,

The curve drawn in Fig. 6.2b is not smooth; it has a sharp turning point dircctly above the point N. Nevertheless, as v tends to N from either side, q again tends to an identical value L. The limit of q again exists and is equal to L.

Figure 6.2c shows what is known as a step function.^ In this case, as v tends to /V, the left-side limit of q is L \, but the right-side limit is ¿2, a different number. Hencc, q docs not have a limit as v N.

Lastly, in Fig. 6.2d, as t1 tends to N, the left-side limit of q is -oo7 whereas the right-side limit is +cc, bccausc the two parts of the (hyperbolic) curve will fall and rise indefinitely while approaching the broken vertical line as an asymptote. Again, lim q does not exist.

On the other hand, if we are considering a different sort of limit in diagram d, namely, lim q, then only the left-side limit has relevance, and wc do find that limit to exist:

lim q = M. Analogously you can verily that lim q = M as well.

It is also possible to apply the conecpts of left-side and right-side limits to the discussion of the marginal cost in Fig, 6.1. In that context, the variables q and u will refer, respectively, to the quotient AC/AQ and to the magnitude of AQ? with all changes being measured from point A on the curve. In other words, q will refer to the slope of such lines as AD, and KG, whereas i; will refer to the length of such lines as Q0Q2 (= line AE) and QoQi (= line A F)t We have already seen that, as v approaches zero from a positive value, q will approach a value equal to the slope of line KG. Similarly we can establish that, if A Q approaches zero from a negative value (i.e., as the decrease in output becomes less and less), the quotient AC/A0, as measured by the slope of such lines as RA (not drawn), will also approach a value equal to the slope of line KG. Indeed, the situation here is very much akin to that illustrated in Fig, 62a. Thus the slope of KG in Fig. 6.1 (the counterpart of L in Fig. 6.2) is indeed the limit of the quotient q as v lends to zero, and as such it gives us the marginal cost at the output level Q = Qq.

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