Section 2.5 The Precise Definition of a Limit
We have worked with limits so far using the intuitive idea that \(\lim_{x \to a} f(x) = L\) means
As \(x\) gets “close” to \(a\text{,}\) \(f(x)\) gets “close” to \(L\text{.}\)
To state this more precisely, we first put this in terms of guarantees of closeness to \(L\text{:}\)
We can guarantee that the value of \(f(x)\) is “close enough” to \(L\) by looking only at values of \(x\) that are “close enough” to \(a\text{.}\)
Next, we measure the closeness of two numbers by the absolute value of their difference being small:
We can guarantee that \(f(x)-L\) is “as small as we want” by looking only at values of \(x\) with \(x-a\) “small enough”.
Finally we give a numerical meaning to “small”:
Guaranteeing \(f(x)-L\) “as small as we want” means smaller than any chosen positive number, \(\epsilon\text{:}\) \(|f(x)-L| < \epsilon\text{;}\)
considering only \(x-a\) “small enough” means only \(x\) values with \(|x-a| < \delta\) for some positive value \(\delta\text{.}\)
Putting this all together gives the precise definition of a limit:
Definition 2.5.1. Limit.
The limit of \(f(x)\) as \(x\) goes to \(a\) is \(L\) if for any given positive number \(\epsilon\text{,}\) there is a positive number \(\delta\) so that having \(|x-a| < \delta\text{,}\) \(x \neq a\) ensures that \(|f(x)-L| < \epsilon\text{.}\)
When this is true, we write \(\lim_{x \to a} f(x) = L.\)
Note that the value of \(f\) at \(x=a\) is ignored, which in particular allows limits to exist even if \(f\) is not defined at \(x=a\text{.}\)
Example 2.5.2.
For \(f(x)=2x+3\text{,}\) \(a=4\text{,}\) verify that the limit is \(L=11\text{:}\) \(\displaystyle \lim_{x \to 4}(2x+3) = 11\text{.}\)
Solution.
This is confirmed by using \(\delta=\epsilon/2\) (this \(\delta\) is positive as required).
This is because for \(|x-4| < \delta\text{,}\) \(|f(x)-11| = |(2x+3)-11| = |2x-8|=|2(x-4)|=2|x-4| < 2\delta = \epsilon\text{.}\) That is, \(|f(x)-11| < \epsilon\text{,}\) as required.
For example, to get \(|f(x)-11| < 0.001\text{,}\) so \(10.999 < 2x+3 < 11.001\text{,}\)
it works to require \(|x-4| < 0.0005\text{,}\) so that “\(x\) is close to \(4\)” in that \(3.9995 < x < 4.0005\text{.}\)
One-sided Limits.
The other types of limits have similar precise definitions. Firstly,
Definition 2.5.3. Right-hand limit.
The right-hand limit of \(f(x)\) as \(x\) goes to \(a\) is \(L\) if for any given positive number \(\epsilon\text{,}\) there is a positive number \(\delta\) so that having \(a < x < a+\delta\) ensures that \(|f(x)-L| < \epsilon\text{.}\) When this is true, we write \(\lim_{x \to a^+} f(x) = L.\)
Checkpoint 2.5.4.
\(\sqrt{x}\) is only defined on one side of \(x=0\text{,}\) so evaluate \(\lim_{x \to 0^+} \sqrt{x}\text{.}\)
Infinite Limits.
We need a slightly different measure for \(f(x)\) being “close to infinity”, and what we use is \(f(x) > M\) for large \(M\text{;}\) likewise having \(f(x) < M\) for large negative \(M\) measures “closeness to \(-\infty\)”.
Definition 2.5.5. Infinite Limit.
The limit of \(f(x)\) as \(x\) goes to \(a\) is infinity if for any given number \(M\text{,}\) there is a positive number \(\delta\) so that
having \(|x-a| < \delta\text{,}\) \(x \neq a\) ensures that \(f(x) > M.\)
When this is true, we write \(\lim_{x \to a} f(x) = \infty.\)
Similarly for \(f\) having a limit of \(-\infty\text{,}\) using \(f(x) < M\) instead.
openstax.org/books/calculus-volume-1/pages/2-5-the-precise-definition-of-a-limit
openstax.org/books/calculus-volume-1/pages/2-5-the-precise-definition-of-a-limit
