Connect a table, graph, and limit statement
One function, three views
The chalk curve shows \(g(x)=\dfrac{\sqrt{x+4}-2}{x}\). Each button moves the orange dot to an input and writes its height as a row of the table under the figure.
The first two buttons move x to the left of 0, where the values 0.2516 and 0.2502 fall toward 0.25 from above.
The last two buttons move x to the right of 0, where the values 0.2498 and 0.2485 rise toward 0.25 from below.
The function has no value at \(x=0\) because the denominator is 0, so the table has no row for that input. The values on both sides approach 0.25, so the limit is 0.25.
Two columns, two sides
The left column lists inputs below \(c\), and the right column lists inputs above \(c\). Each column gives the values of \(f\) as \(x\) comes close to \(c\) from one side.
The slider adds rows, and each new row has an input closer to \(c\). Under each column, an orange arrow points to the number that the values approach.
This number is a one-sided limit, the left-hand limit for the left column and the right-hand limit for the right column.
If both columns approach the same number \(L\), the graph approaches height \(L\) from both sides. No row uses the value at \(c\).
Notation
The left column gives the left-hand limit, where the minus sign means that \(x\) is below \(c\).
\[\lim_{x\to c^-} f(x)=\cy{L_1}\]The right column gives the right-hand limit, where the plus sign means that \(x\) is above \(c\).
\[\lim_{x\to c^+} f(x)=\cy{L_2}\]If the two one-sided limits are equal, the limit exists and has the same value, because \(f(x)\) approaches \(L\) from either side.
\[\begin{gathered}\lim_{x\to c} f(x)=\cy{L}\\ \text{when}\ L_1=L_2=L\end{gathered}\]A table shows only a few inputs, and a graph shows heights only roughly, so both suggest a limit. The formula confirms the value.
When it fails
The left side approaches the same height as before, and the right side approaches a height 1 higher.
The one-sided limits differ, so no single height fits both sides. The limit at \(c\) does not exist, although each one-sided limit does.
Example 1: two columns that disagree
Let \(p(x)=2x+1\) for \(x<3\), \(p(x)=12-x\) for \(x>3\), and \(p(3)=5\). Find the limit of \(p\) at 3.
Build the table from inputs close to 3 on each side, but not from 3 itself.
\[\begin{array}{c|cc|cc} x & 2.9 & 2.99 & 3.01 & 3.1\\ \hline p(x) & 6.8 & 6.98 & 8.99 & 8.9 \end{array}\]The left column approaches 7, because the line \(2x+1\) is at height 7 when \(x=3\).
\[\lim_{x\to 3^-} p(x)=\cy{7}\]The right column approaches 9, because the line \(12-x\) is at height 9 when \(x=3\).
\[\lim_{x\to 3^+} p(x)=\cy{9}\]The one-sided limits are different, so the limit at 3 does not exist. The dot at \((3,5)\) shows the value \(p(3)\), and neither column uses it.
Example 2: a dot away from the curve
Let \(u(x)=2x-3\) for \(x\neq 4\), and \(u(4)=9\). Find the limit of \(u\) at 4.
Build the table from inputs close to 4 on each side, but not from 4 itself.
\[\begin{array}{c|cc|cc} x & 3.9 & 3.99 & 4.01 & 4.1\\ \hline u(x) & 4.8 & 4.98 & 5.02 & 5.2 \end{array}\]A tempting wrong move: The graph has a dot at height 9 when \(x=4\), so the limit is 9.
\[\lim_{x\to 4} u(x)=\cr{9}\]The dot at height 9 is not on the line. On both sides of 4 the line is close to height 5, far below the red line.
A limit uses the inputs near 4, and not the input 4. Both columns approach 5, so the limit is 5. The value \(u(4)=9\) is a separate fact.
\[\lim_{x\to 4} u(x)=\cy{5}\]
Practice
A table for \(v\) near \(x=1\) lists \(v(0.9)=2.7\), \(v(0.99)=2.97\), \(v(0.999)=2.997\), \(v(1.001)=3.003\), \(v(1.01)=3.03\) and \(v(1.1)=3.3\). Also \(v(1)=7\). Write the two one-sided limits and the limit at 1.
Read each column on its own. Then compare the two heights. The value v(1) is in neither column.
Solution
The values in the left column rise toward 3.
\[\lim_{x\to 1^-} v(x)=\cy{3}\]The values in the right column fall toward 3.
\[\lim_{x\to 1^+} v(x)=\cy{3}\]The one-sided limits are equal, so the limit is 3. The value \(v(1)=7\) does not change it.
\[\lim_{x\to 1} v(x)=\cy{3}\]
Questions
The graph shows a function \(f\). At \(x=-2\) it has an open ring and a separate filled dot. At \(x=3\) the curve jumps.
As \(x\) approaches \(-2\), which height does \(f\) approach?
- 4
- 1
- The limit does not exist, because the ring and the dot differ
- \(-2\)
Answer
4
What is the right-hand limit of \(f\) as \(x\) approaches 3?
- 5
- 9
- It does not exist, because the curve jumps
- 3
Answer
5
Let \(h(x)=2x+3\) for \(x<1\) and \(h(x)=9-4x\) for \(x\geq 1\). Find \(\displaystyle\lim_{x\to 1} h(x)\).
- 5
- The limit does not exist, because h has two formulas
- 9
- 3
Answer
5
Summary
You read each column of a table as one side of the graph, and you write it as a one-sided limit.
This works because a limit describes the values near the input from both sides, and not the value at the input.
It lets you move between a table, a graph and a limit statement, and it shows when the two sides disagree.
This is one of four skills in topic 1.9 of Which & Why: AP Calculus.