Lesson 1.2 · Unit 1 · CED topic 1.2
Defining limits and using limit notation
Lesson 1.1 ended with a promise: instantaneous rates of change are defined by what average rates "approach." Now we make approach precise. A limit describes the value a function gets close to as its input gets close to some number. Limits are the language the rest of calculus is written in, so it is worth learning the notation until it reads like a sentence.
Learning objectives.
- LIM-1.A. Represent limits analytically using correct notation.
- LIM-1.B. Interpret limits expressed in analytic notation.
- LIM-1.C. Estimate limits of functions.
Essential knowledge.
- LIM-1.A.1. Given a function \(f\), the limit of \(f(x)\) as \(x\) approaches \(c\) is a real number \(R\) if \(f(x)\) can be made arbitrarily close to \(R\) by taking \(x\) sufficiently close to \(c\) (but not equal to \(c\)). If the limit exists and is a real number, then the common notation is \(\lim_{x\to c}f(x)=R\).
- LIM-1.B.1. A limit can be expressed in multiple ways, including graphically, numerically, and analytically.
- LIM-1.C.1. The concept of a limit includes one-sided limits.
The idea of a limit
Suppose \(f\) is a function and \(c\) is a number. We say that the limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\), and write
\[ \lim_{x\to c} f(x)=L, \]
when the values \(f(x)\) can be made as close to \(L\) as we like by taking \(x\) close enough to \(c\), but not equal to \(c\). The words "but not equal to \(c\)" matter. A limit is about what happens near \(c\), not at \(c\). That is exactly what we needed in Lesson 1.1: a quotient that is undefined at \(h=0\) can still have a perfectly good limit as \(h\) approaches \(0\).
You read the notation left to right: "the limit, as \(x\) approaches \(c\), of \(f\) of \(x\), equals \(L\)." A common shorthand is \(f(x)\to L\) as \(x\to c\). The number \(L\) is a real number if the limit exists; later (Lesson 1.14) we will also use the symbol \(\infty\) to describe certain behavior, but \(\infty\) is not a number and a limit that "equals infinity" is a limit that does not exist as a real number.
The figure shows why the value at the point and the limit at the point are separate things. The graph follows the line \(y=x+1\) everywhere except \(x=1\). Near \(x=1\) the outputs head toward \(2\), so \(\lim_{x\to1}f(x)=2\). The function value happens to be \(f(1)=0.5\), but the limit does not care. The function could even be undefined at \(x=1\) and the limit would still be \(2\).
One-sided limits
Sometimes the left and right sides of \(c\) behave differently, so we need notation for each side. The left-hand limit \(\displaystyle\lim_{x\to c^-}f(x)\) uses only inputs \(x\lt c\), and the right-hand limit \(\displaystyle\lim_{x\to c^+}f(x)\) uses only inputs \(x\gt c\). The small superscript sign tells you which side you approach from: minus for smaller values (the left on a number line), plus for larger values (the right).
The two-sided limit and the one-sided limits are related by one rule that you will use constantly:
\[ \lim_{x\to c}f(x)=L \quad\text{exactly when}\quad \lim_{x\to c^-}f(x)=L \ \text{ and }\ \lim_{x\to c^+}f(x)=L. \]
In words: the two-sided limit exists only when both one-sided limits exist and agree. If they exist but differ, the two-sided limit does not exist.
Let \(f(x)=\begin{cases}x+2, & x\lt1\\ 1, & x=1\\ 5-x, & x\gt1.\end{cases}\) Find \(\displaystyle\lim_{x\to1^-}f(x)\), \(\displaystyle\lim_{x\to1^+}f(x)\), \(\displaystyle\lim_{x\to1}f(x)\), and \(f(1)\).
Left. For \(x\lt1\), \(f(x)=x+2\), and \(x+2\to3\) as \(x\to1\), so the left-hand limit is \(3\).
Right. For \(x\gt1\), \(f(x)=5-x\), and \(5-x\to4\), so the right-hand limit is \(4\).
Two-sided. The one-sided limits are different (\(3\ne4\)), so \(\displaystyle\lim_{x\to1}f(x)\) does not exist.
Value. \(f(1)=1\), which agrees with neither limit. The function value and the limits are three separate pieces of information.
The same limit in three representations
A limit can be shown graphically (follow the curve toward \(x=c\) from each side and read the height), numerically (a table of values for \(x\) near \(c\)), and analytically (with a formula and algebra). All three describe the same number. In this lesson you read limits from graphs and tables and write them correctly; algebraic evaluation comes in Lessons 1.5 and 1.6.
Estimate \(\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}\) from a table.
| \(x\) | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
|---|---|---|---|---|---|---|
| \(\dfrac{x^2-4}{x-2}\) | 3.9 | 3.99 | 3.999 | 4.001 | 4.01 | 4.1 |
The function is undefined at \(x=2\) (it gives \(\frac{0}{0}\)), yet the values from both sides close in on \(4\). So \(\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}=4\). The table is evidence, not a proof, but here the pattern is clear and an algebraic check agrees: \(\dfrac{x^2-4}{x-2}=\dfrac{(x-2)(x+2)}{x-2}=x+2\) for \(x\ne2\).
- \(\displaystyle\lim_{x\to c}f(x)=L\) means \(f(x)\) is close to \(L\) when \(x\) is close to, but not equal to, \(c\).
- The limit does not depend on \(f(c)\). \(f(c)\) may equal the limit, differ from it, or not exist.
- \(c^-\) means approach from the left (smaller \(x\)); \(c^+\) means approach from the right (larger \(x\)).
- The two-sided limit exists only if both one-sided limits exist and are equal. If they differ, write "does not exist" (not "0" and not "undefined").
- Write limit notation completely, with the variable and the value approached, every line you use it. Dropping the "\(\lim\)" before the limit is reached is a classic lost point.
- Read aloud and rewrite in words: \(\displaystyle\lim_{x\to3^-}g(x)=5\).
Show answer
"The limit of \(g\) of \(x\) as \(x\) approaches 3 from the left is 5." The outputs of \(g\) get close to 5 as \(x\) approaches 3 through values less than 3.
- Use limit notation for this statement: as \(x\) approaches 4 from the right, \(f(x)\) gets close to 6.
Show answer
\(\displaystyle\lim_{x\to4^+}f(x)=6\).
- A graph shows an open circle at \((2,3)\) and a closed circle at \((2,7)\), with the curve approaching height 3 from both sides of \(x=2\). Find \(\displaystyle\lim_{x\to2}f(x)\) and \(f(2)\).
Show answer
The curve approaches height 3 from both sides, so the limit is 3. The closed circle gives the function value, \(f(2)=7\). The limit and the function value differ.
- For \(h(x)=\dfrac{|x|}{x}\), find \(\displaystyle\lim_{x\to0^-}h(x)\), \(\displaystyle\lim_{x\to0^+}h(x)\), and \(\displaystyle\lim_{x\to0}h(x)\).
Show answer
For \(x\lt0\), \(|x|=-x\), so \(h(x)=-1\); for \(x\gt0\), \(h(x)=1\). The left limit is \(-1\), the right limit is \(1\), and since they differ, \(\displaystyle\lim_{x\to0}h(x)\) does not exist.
- The values of \(f(x)=\dfrac{x^3-1}{x-1}\) are \(f(0.9)=2.71\), \(f(0.99)=2.9701\), \(f(0.999)=2.997001\), and \(f(1.001)=3.003001\). Estimate \(\displaystyle\lim_{x\to1}f(x)\). Is \(f(1)\) defined?
Show answer
The values from both sides approach 3, so the limit is approximately 3. \(f(1)\) is not defined, because the formula gives \(\frac{0}{0}\). The limit exists anyway.
- True or false: if \(f(c)\) is undefined, then \(\displaystyle\lim_{x\to c}f(x)\) does not exist. Explain.
Show answer
False. A limit depends only on values near \(c\), not at \(c\). For example \(\dfrac{x^2-4}{x-2}\) is undefined at \(x=2\) but has limit 4 there.