Read the value at the point
Three graphs, one curve
Press Slide the point toward 2. The orange point moves along the curve and its height comes closer to 3, so the limit at 2 is 3. The open ring at height 3 marks a point that the graph leaves out.
Press No dot. The graph has nothing at \(x = 2\), so \(f(2)\) is undefined. The limit is still 3.
Press Dot below the ring. A filled dot is a point that the graph includes, and this one is at height 1, so \(f(2) = 1\) while the limit is still 3.
Press Dot on the curve. The dot fills the ring, so \(f(2) = 3\). Only here does the value equal the limit.
Any curve, any dot
Without numbers, the curve heads toward a height \(\cy{L}\) as \(x\) comes close to \(c\), and the orange open ring marks that height above \(c\).
Drag the Dot height slider. The height of the filled dot is the value \(f(c)\), and the limit \(\cy{L}\) does not move.
Press Remove the dot. The function now has no value at \(c\), but the limit is still \(L\).
The limit depends on the curve near the input. The value depends only on the dot at the input.
Notation
The value of the function at \(c\) is the height of the filled dot above the input \(c\).
\[f(c)\]If no filled dot stands at \(c\), then \(f(c)\) is undefined.
The limit at \(c\) is the height that the curve approaches as \(x\) comes close to \(c\), so it ignores the point at \(c\) itself.
\[\lim_{x \to c} f(x) = \cy{L}\]There are three cases: no filled dot, a filled dot off the curve, or a filled dot on the curve. The value equals the limit only in the last case.
When it fails
Press Slide the dot off the curve. The value \(f(c)\) changes, but the limit \(\cy{L}\) stays the same, because the limit uses only inputs near \(c\), not \(c\) itself.
If you read the limit from a dot that is off the curve, your answer is wrong. Read the limit from the curve and the value from the dot.
Example 1: a dot off the curve
The graph of \(f\) is the curve \(y = 5 - x^2\), with an open ring at \((2, 1)\) and a filled dot at \((2, 4)\). Find \(f(2)\) and \(\lim_{x \to 2} f(x)\).
The value is the height of the filled dot, which stands at 4 above \(x = 2\).
\[f(2) = \cy{4}\]For the limit, ignore the dot and follow the curve. At \(x = 1.99\) the height is \(5 - 1.99^2 = 1.0399\).
The heights approach the open ring, at \(5 - 2^2 = 1\).
\[\lim_{x \to 2} f(x) = \cy{1}\]The value and the limit are different, because the dot is off the curve.
Example 2: no dot
Let \(h(x) = \dfrac{x^2 - x - 12}{x - 4}\). Its graph is the line \(y = x + 3\), with an open ring at \((4, 7)\) and no filled dot. Find \(h(4)\) and \(\lim_{x \to 4} h(x)\).
The numerator factors as \((x - 4)(x + 3)\). For \(x \neq 4\), you can cancel \(x - 4\) because it is not 0, so the quotient is \(x + 3\). At \(x = 3.99\) the height is 6.99.
\[x^2 - x - 12 = (x - 4)(x + 3)\]The heights approach 7, so the limit is 7.
\[\lim_{x \to 4} h(x) = \cy{7}\]A tempting wrong move: The ring is at height 7, so the value is 7.
\[h(4) = 7\]The red cross marks an error. An open ring is a point that the graph leaves out, and no filled dot stands above x = 4, so h has no value there. The ring shows only where the line heads.
At \(x = 4\) the quotient is \(\tfrac{0}{0}\), which is not a number, so \(h(4)\) is undefined. The limit is still 7.
\[h(4) \text{ is undefined}\]
Practice
The graph of \(p\) is the line \(y = 2x + 3\), with a filled dot on the line at \(x = 2\). Find \(p(2)\) and \(\lim_{x \to 2} p(x)\). Are they equal?
Read the value from the filled dot. Read the limit from the line near 2.
Solution
The dot is at height 7, so \(p(2) = 7\).
\[p(2) = \cy{7}\]The heights on the line approach \(2(2) + 3 = 7\).
\[\lim_{x \to 2} p(x) = \cy{7}\]The value equals the limit, because the dot stands on the line.
Questions
Use the graph for the first two questions.
What is \(f(-2)\)?
- \(4\)
- \(1\)
- \(f(-2)\) is undefined.
Answer
\(4\)
What is \(\lim_{x \to 1} f(x)\)?
- \(4\)
- The limit does not exist, because \(f(1)\) is undefined.
- \(1\)
Answer
\(4\)
Let \(g(x) = x^2 + 1\) for \(x \neq 3\), and let \(g(3) = 20\). What is \(\lim_{x \to 3} g(x)\)?
- \(10\)
- \(20\)
- The limit does not exist.
Answer
\(10\)
Summary
You read the value from the filled dot at the input, and you read the limit from the height that the curve approaches.
This works because the limit ignores the point at the input, and the value ignores the curve around it.
Use it when a graph shows an open ring, a filled dot, or both at the same input.
This is one of four skills in topic 1.3 of Which & Why: AP Calculus.