Chapter 1: Limits
Section 1.1: Naive Limits
Example 1.1.2
Using a graph and a table of values, estimate limx→0 gx, where gx=100 x−cosx2100000.
Solution
Control-drag (or type) the equation gx=…
Context Panel: Assign Function
gx=100 x−cosx2100000→assign as functiong
From the graph of gx in Figure 1.1.2(a), one might conclude that limx→0gx=0.
This conclusion might be reinforced by the data in Table 1.1.2(a) where it appears that as x approaches zero from either side, the values of gx seem to tend towards zero.
Figure 1.1.2(a) Graph of gx on the interval −1,1
a := 0: H := [ 1.0, 0.5, 0.1, 0.01 ]: xL := [seq( a-h, h=H )]: xR := [seq( a+h, h=H )]: header := < `x` | `g(x)` >, < `_____` | `_______________` >: bodyL := seq( < X | g(X) >, X= a-H ): tableL := < header, bodyL >: bodyR := seq( < X | g(X) >, X= a+H ): tableR := < header, bodyR >: tableL, tableR;
Table 1.1.2(a) Numeric estimate of limx→0 gx
Expression palette: Limit operator
Context Panel: Evaluate and Display Inline
limx→0gx = 1100000
Applying Maple's limit operator to gx shows that the actual limit as x→0 is not zero, but a small positive number.
The graph of gx in Figure 1.1.2(b), drawn on a smaller interval about x=0, also shows that the limit is not zero, but the small positive number 10−5.
Table 1.1.2(b) evaluates gx at points closer to x=0.
Figure 1.1.2(b) Graph of gx,x∈0,1/50
a := 0: H := [0.01,.001,.0001,.00001,.000001 ]: xL := [seq( a-h, h=H )]: xR := [seq( a+h, h=H )]: header := < `x` | `g(x)` >, < `_____` | `_______________` >: bodyL := seq( < X | g(X) >, X= a-H ): tableL := < header, bodyL >: bodyR := seq( < X | g(X) >, X= a+H ): tableR := < header, bodyR >: tableL, tableR;
Table 1.1.2(b) Finer table of values for gx near x=0
Based on Figure 1.1.2(a) and Table 1.1.2(a), one might conclude that limx→0gx=0. Applying Maple's built-in limit operator shows that the limit is actually the small positive number 10−5. The potential for error in relying solely on graphs and tables is significant. However, Figure 1.1.2(b) and Table 1.1.2(b) do show that with proper care and attention, it is possible to deduce the value of a limit from a graph or table of values. But it is well worth learning the more efficient analytic methods of calculus.
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