Lesson 1.3 · Unit 1 · CED topics 1.3 and 1.4
Estimating limit values from graphs and tables
You will often be handed a limit with no formula to work with: a graph in a free-response question, or a table of measured data. This lesson is about reading limits from those two representations, and just as important, about knowing when they can fool you. Algebraic techniques come next; first you should be able to look at a picture or a column of numbers and say what a limit is, or why it does not exist.
Learning objective.
- LIM-1.C. Estimate limits of functions.
Essential knowledge.
- LIM-1.C.2. Graphical information about a function can be used to estimate limits.
- LIM-1.C.3. Because of issues of scale, graphical representations of functions may miss important function behavior.
- LIM-1.C.4. A limit might not exist for some functions at particular values of \(x\). Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.
- LIM-1.C.5. Numerical information can be used to estimate limits.
Estimating a limit from a graph
To find \(\displaystyle\lim_{x\to c}f(x)\) from a graph, put one finger on the curve far to the left of \(x=c\) and slide it toward \(x=c\); note the height it approaches. Then do the same from the right. If both fingers head for the same height \(L\), then \(\displaystyle\lim_{x\to c}f(x)=L\). Three things to remember:
- An open circle marks a point the graph approaches but does not include; a closed circle marks the actual function value. The limit is the height the curve approaches, whatever the circles say.
- If the two sides approach different heights, the limit does not exist, even though each one-sided limit does.
- The function value \(f(c)\) is read separately from the filled-in point. It may differ from the limit.
The graph of \(g\) has these features. Near \(x=-1\) the curve approaches height 2 from both sides, with a closed circle at \((-1,4)\). At \(x=3\) the curve approaches height 1 from the left and height \(-1\) from the right, with a closed circle at \((3,0)\). Find the requested values.
| Quantity | Value | Reason |
|---|---|---|
| \(\displaystyle\lim_{x\to-1}g(x)\) | 2 | both sides approach 2 |
| \(g(-1)\) | 4 | the closed circle |
| \(\displaystyle\lim_{x\to3^-}g(x)\) | 1 | left side approaches 1 |
| \(\displaystyle\lim_{x\to3^+}g(x)\) | \(-1\) | right side approaches \(-1\) |
| \(\displaystyle\lim_{x\to3}g(x)\) | does not exist | the one-sided limits differ |
| \(g(3)\) | 0 | the closed circle |
Three ways a limit can fail to exist
A limit might not exist for some functions at particular values of \(x\). There are three common reasons, and you should be able to name them from a graph or a formula.
- The one-sided limits disagree (a jump). Example: the function in the figure's left panel, or \(\dfrac{|x|}{x}\) at \(0\).
- The function is unbounded near \(c\) (it grows without bound). Example: \(\dfrac{1}{x^2}\) as \(x\to0\). The outputs do not settle on any real number.
- The function oscillates near \(c\) without settling. Example: \(\sin\dfrac{1}{x}\) as \(x\to0\), which swings between \(-1\) and \(1\) infinitely often in every interval around \(0\).
(Notice that a limit that does not exist is a perfectly good mathematical answer. On the exam, "does not exist" with a reason is often the whole point of the question.)
Estimating a limit from a table
A table of values gives numerical information about a limit. Choose \(x\)-values that approach \(c\) from both sides, closer and closer, and watch the outputs. If the outputs from both sides approach the same number, that number is your estimate of the limit.
Estimate \(\displaystyle\lim_{x\to0}\frac{\sin x}{x}\) (with \(x\) in radians).
| \(x\) | \(-0.1\) | \(-0.01\) | \(-0.001\) | 0.001 | 0.01 | 0.1 |
|---|---|---|---|---|---|---|
| \(\dfrac{\sin x}{x}\) | 0.998334 | 0.999983 | 0.9999998 | 0.9999998 | 0.999983 | 0.998334 |
The function is not defined at \(x=0\), but the values from both sides climb toward \(1\). We estimate \(\displaystyle\lim_{x\to0}\dfrac{\sin x}{x}=1\). (Lesson 1.6 and Lesson 4.7 give ways to prove it.)
For a function \(f\), a table gives \(f(1.9)=4.9\), \(f(1.99)=4.99\), \(f(1.999)=4.999\), and \(f(2.001)=1.001\), \(f(2.01)=1.01\), \(f(2.1)=1.1\). What is \(\displaystyle\lim_{x\to2}f(x)\)?
From the left the values approach \(5\); from the right they approach \(1\). The one-sided limits exist but differ, so \(\displaystyle\lim_{x\to2}f(x)\) does not exist. A table that shows only one side of \(c\) cannot tell you this, which is why a good table always includes both sides.
When graphs and tables mislead
Because limits describe what happens arbitrarily close to \(c\), pictures and finite tables can mislead:
- Scale. A graphing window shows only finite detail. A tiny hole or a very narrow spike may not be visible, and a graph can appear to touch an asymptote. When a limit decision matters, zoom in or use algebra.
- Unlucky inputs. For \(f(x)=\sin\dfrac{\pi}{x}\), the inputs \(x=0.5,\,0.25,\,0.125,\dots\) all give \(\sin(2\pi),\sin(4\pi),\sin(8\pi)=0\), so that table suggests a limit of \(0\). But at \(x=0.4\), \(0.2222\dots\), \(0.1538\dots\) (that is, \(x=\frac{2}{4n+1}\)) the value is \(1\). The function oscillates and the limit at \(0\) does not exist.
- Too few values. Two or three nearby inputs are a suggestion. Use inputs that get steadily closer, from both sides.
A table or graph can estimate a limit; to justify one on the free-response section you will use algebra, limit properties, or a named theorem.
A graphing calculator is valuable for limit work: graph the function and zoom in at \(x=c\), or use a table (TABLE or TBLSET) with a small step. Remember that the calculator evaluates at specific numbers only. If your table or window disagrees with an algebraic argument, trust the algebra and then look for why the picture misled.
- Read a limit from a graph by following the curve from both sides; circles show values, not limits.
- A limit fails to exist when the sides disagree, when the function is unbounded, or when it oscillates.
- For a table, use inputs approaching \(c\) from both sides; agreement suggests a limit, disagreement shows it does not exist.
- Graphs and tables are estimates: scale and unlucky inputs can mislead, so a justification should use algebra or a theorem.
- A table shows \(f(2.9)=7.41\), \(f(2.99)=7.9401\), \(f(2.999)=7.994001\), \(f(3.1)=8.61\), \(f(3.01)=8.0601\), \(f(3.001)=8.006001\). Estimate \(\displaystyle\lim_{x\to3}f(x)\).
Show answer
The values from both sides close in on 8, so \(\displaystyle\lim_{x\to3}f(x)\approx8\). (These are the values of \(x^2-1\).)
- For \(f(x)=\dfrac{1-\cos x}{x^2}\), the table gives \(f(\pm0.1)=0.4995835\) and \(f(\pm0.01)=0.4999958\). Estimate \(\displaystyle\lim_{x\to0}f(x)\).
Show answer
The values approach \(0.5\) from both sides, so the limit is approximately \(0.5\), or \(\frac{1}{2}\).
- A graph shows a curve climbing without bound on both sides of \(x=2\). Does \(\displaystyle\lim_{x\to2}f(x)\) exist? Why?
Show answer
It does not exist as a real number, because the function values are unbounded near \(x=2\) and do not approach any single number.
- Explain why the table \(x=0.5,\,0.25,\,0.125\) for \(f(x)=\sin\dfrac{\pi}{x}\) does not prove that the limit as \(x\to0\) is 0. Find an input near 0 where \(f\) equals 1.
Show answer
All three inputs give \(\sin\) of a multiple of \(\pi\), which is 0, but the function oscillates. At \(x=0.4\), \(f(0.4)=\sin(2.5\pi)=1\). Inputs of the form \(\dfrac{2}{4n+1}\) give 1 for every \(n\), however close to 0, so the limit does not exist.
- On a graph, \(f\) approaches 1 as \(x\to4\) from the left and approaches 3 as \(x\to4\) from the right, and \(f(4)=2\). Find \(\displaystyle\lim_{x\to4^-}f(x)\), \(\displaystyle\lim_{x\to4^+}f(x)\), \(\displaystyle\lim_{x\to4}f(x)\), and \(f(4)\).
Show answer
The left limit is 1, the right limit is 3, the two-sided limit does not exist (the sides disagree), and \(f(4)=2\).
- A calculator graph of \(y=\dfrac{x^2-9}{x-3}\) in a standard window looks like the straight line \(y=x+3\) with no break. What does the graph hide, and what is \(\displaystyle\lim_{x\to3}\dfrac{x^2-9}{x-3}\)?
Show answer
The function is undefined at \(x=3\), so the true graph has a hole at \((3,6)\) that the pixels do not show. Since \(\dfrac{x^2-9}{x-3}=x+3\) for \(x\ne3\), the limit is 6.