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Chapter 1: Limits
Section 1.1: Naive Limits
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Example 1.1.1
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Graph and thereby estimate and .
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Solution
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Context Panel: Assign Function
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The transparent green dot on the graph in Figure 1.1.1(a) is at the point , corresponding to the value = . If evaluation produces a real number, that real number is the limit. Hence,
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=
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The red circle at indicates a gap in the function at because is undefined: a division-by-zero error occurs if the function is evaluated at . However, the graph suggests the estimate
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=
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Figure 1.1.1a Graph of and the points and
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Table 1.1.1(a) lists values of to either side of . These values are consistent with Figure 1.1.1(a), and with the estimate of the limit at .
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a := 4:
H := [ 0.5, 0.1, 0.01, 0.001 ]:
xL := [seq( a-h, h=H )]:
xR := [seq( a+h, h=H )]:
header := < `x` | `f(x)` >,
< `_____` | `_______________` >:
bodyL := seq( < X | f(X) >, X= xL ):
tableL := < header, bodyL >:
bodyR := seq( < X | f(X) >, X= xR ):
tableR := < header, bodyR >:
tableL, tableR;
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Table 1.1.1(a) Numeric estimate of
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