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Activity2.5Limits at Infinity Tending to Zero

When using limit laws to establish limit values as \(x\to\infty\) or \(x\to-\infty\), Limit Law A1Limit Law A6 and Limit Law R2 are still in play (when applied in a valid manner), but Limit Law R1 cannot be applied. (The reason it cannot be applied is discussed in detail in Exercise Group 2.8.1.11–2.8.1.18 from Activity 2.8.)

There is a new replacement law that can only be applied when \(x\to\infty\) or \(x\to-\infty\); this is Limit Law R3. Limit Law R3 essentially says that if the value of a function is increasing without any bound on how large it becomes or if the function is decreasing without any bound on how large its absolute value becomes, then the value of a constant divided by that function must be approaching zero. An analogy can be found in extremely poor party planning. Let's say that you plan to have a pizza party and you buy five pizzas. Suppose that as the hour of the party approaches more and more guests come in the door—in fact the guests never stop coming! Clearly as the number of guests continues to rise the amount of pizza each guest will receive quickly approaches zero (assuming the pizzas are equally divided among the guests).

Subsection2.5.1Exercises

1

Consider the function \(f\) defined by \(\fe{f}{x}=\frac{12}{x}\).

Complete Table 2.5.1 without the use of your calculator. What limit value and limit law are being illustrated in the table?

\(x\) \(\fe{f}{x}\)
\(1000\)
\(10{,}000\)
\(100{,}000\)
\(1{,}000{,}000\) \(\phantom{1{,}000{,}000}\)
Table2.5.1\(\fe{f}{x}=\frac{12}{x}\)