Theorem 1.3.1.
If \(f(x)\) is constant, say \(f(x)=k\) for some number \(k\text{,}\) and \(a\) is any number, Then \(\displaystyle \lim_{x\to a}f(x)=k\text{.}\)
Section 1.3 Properties
A careful observer will notice that in the previous sections, we only looked at a handful of domain values to essentially make a guess as to where the function is headed. Consider, for example, our friend \(sin(x)/x\text{.}\) Look at the table we made at the beginning of the first section. We stopped looking once we let \(x\) get down to \(0.001\text{.}\) But what about even closer? Like \(x=0.0003\) or \(x=0.0000000162\) or \(x=0.0000000000004\text{?}\) Maybe waaaay down there \(sin(x)/x\) starts getting wonky. Maybe. So, right now we’re just guessing that \(\displaystyle \lim_{x\to 0^{+}}(sin(x)/x)=1\text{.}\) Educated guessing ... yes. But still just guessing. We’d like to start moving past just guessing. We start doing that in this section. We start by looking at a couple of functions whose limits are not in question.
First, consider a constant function like \(f(x)=5\text{.}\) No matter what you put in for \(x\text{,}\) it gives us \(5\text{.}\) So, there’s no question what it’s approaching regardless of where we let \(x\) go. It’s always giving \(5\text{.}\) So, we have ...
Secondly, consider the function \(f(x)=x\text{.}\) Well, this function just does whatever \(x\) does. So, where \(x\) goes, so does \(f(x)=x\text{.}\) Thus, we have ...
Theorem 1.3.2.
It is our good fortune that limits behave according to the following theorem.
Theorem 1.3.3.
If \(\displaystyle \lim_{x\to a}f(x)=l\) and \(\displaystyle \lim_{x\to a}g(x)=k\) and \(r\) and \(s\) are any two numbers and the right hand side of the following equalities are defined, Then \(\displaystyle \lim_{x\to a}(rf(x)+sg(x))=rl+sk\text{,}\) \(\displaystyle \lim_{x\to a}(f(x)g(x))=lk\text{,}\) and \(\displaystyle \lim_{x\to a}(f(x)/g(x))=l/k\text{.}\)
Remark 1.3.4.
\(\displaystyle \lim_{x\to a}\)\(\displaystyle \lim_{x\to a^{+}}\)\(\displaystyle \lim_{x\to a^{-}}\text{,}\)Theorem 1.3.3Often, people cite this basic properties theorem by saying, "the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a ratio is the ratio of the limits".
Remark 1.3.5.
\(k=0\)\(\displaystyle \lim_{x\to a}(f(x)/g(x))=l/k\)\(l/0\)
Example 1.3.6.
Using Theorem 1.3.3, we quickly deduce the following quick conclusions (called "corollaries" by mathematicians).
Corollary 1.3.7.
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If \(q\) is a nonzero rational number and \(a\) is a nonzero number at which \(a^q\) is defined, then \(\displaystyle \lim_{x\to a}x^q=a^q\text{.}\)
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If \(p(x)\) and \(q(x)\) are polynomials and \(q(a)\ne 0\text{,}\) then \(\displaystyle \lim_{x\to a}(p(x)/q(x))=p(a)/q(a)\text{.}\)
Finally, we provide here a short list of known limits for use in subsequent sections.
