Calculus BC ยท Unit 1
Read one-sided limits, compositions, and combinations from a graph or a table.
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Read y as x approaches
The limit is the y-value the graph is heading toward, not the filled point. An open circle can still have a limit.
One-sided limits
\(x \to a^-\) is from the left. \(x \to a^+\) is from the right. Both sides must match for the two-sided limit.
Composition on a graph
For \(\lim f(f(x))\), first see what f(x) approaches. Then take that one-sided limit on the graph of f.
Tables use limits, not f(a)
When a table gives both \(f(a)\) and \(\lim f\), use the limits in a limit calculation. The function value is only for continuity.
Read a one-sided or two-sided limit from the graph.
Graph of f
From the left, the graph heads toward the open circle at \((-1, 4)\).
\(\lim_{x \to -1^-} f(x) = 4\)
Answer: \(4\)
Graph of f
From the right, the graph starts at the filled point \((-1, 2)\) and stays at height 2.
\(\lim_{x \to -1^+} f(x) = 2\)
Answer: \(2\)
Graph of f
\(\lim_{x \to -1^-} f(x) = 4\)
\(\lim_{x \to -1^+} f(x) = 2\)
The one-sided limits are not equal.
does not exist
Answer: does not exist
Graph of f
The filled point at \(x = -1\) is at height 2.
\(f(-1) = 2\)
Answer: \(2\)
Graph of f
Near \(x = 1\) the graph is the horizontal piece at height 2, and the filled point is on that piece.
\(\lim_{x \to 1} f(x) = 2\)
Answer: \(2\)
Graph of f
From the left the graph heads toward the open circle at \((3, 4)\).
\(\lim_{x \to 3^-} f(x) = 4\)
Answer: \(4\)
Composition \(f(f(x))\), or a table of limits.
Graph of f
As \(x \to -1^-\), \(f(x)\) approaches 4 from below.
\(\lim_{x \to -1^-} f\bigl(f(x)\bigr) = \lim_{y \to 4^-} f(y)\)
On the graph, near \(x = 4\) the last piece has height 0 at \(x = 4\).
\(\lim_{y \to 4^-} f(y) = 0\)
Answer: \(0\)
Graph of f
As \(x \to -1^+\), \(f(x)\) approaches 2.
\(\lim_{x \to -1^+} f\bigl(f(x)\bigr) = \lim_{y \to 2} f(y)\)
Near \(x = 2\) the middle piece goes from \((1, 2)\) to \((3, 4)\), so the height at 2 is 3.
\(\lim_{y \to 2} f(y) = 3\)
Answer: \(3\)
Graph of f
Left-hand composition limit is 0.
Right-hand composition limit is 3.
The one-sided limits are not equal.
does not exist
Answer: does not exist
Graph of f
As \(x \to 0^-\), \(f(x)\) approaches 3 from below.
\(\lim_{x \to 0^-} f\bigl(f(x)\bigr) = \lim_{y \to 3^-} f(y)\)
Near \(x = 3\) the middle piece goes from \((2, 3)\) to \((4, 5)\), so the height at 3 is 4.
\(4\)
Answer: \(4\)
Use the limits, not the function values.
\(\lim_{x \to 1}\bigl(2f(x) + g(x)\bigr) = 2\lim f + \lim g\)
\(2(2) + 3\)
\(7\)
Answer: \(7\)
\(\lim h \cdot \bigl(\lim f + 3\lim g\bigr)\)
\(2\bigl(4 + 3(-1)\bigr)\)
\(2(4 - 3)\)
\(2(1)\)
\(2\)
Answer: \(2\)
A graph plus a table, or a composition through a vertical asymptote.
Graph of f
| \(x\) | 3.8 | 3.9 | 3.99 | 4 | 4.01 | 4.1 | 4.2 |
|---|---|---|---|---|---|---|---|
| \(g(x)\) | 3.41 | 3.70 | 3.97 | 4.00 | 4.03 | 4.31 | 4.62 |
From the graph, \(f(4) = 2\) and \(f\) is continuous at 4, so \(\lim_{x \to 4} f(x) = 2\).
From the table, \(g(x)\) approaches 4 as \(x \to 4\).
\(\lim_{x \to 4}\bigl(2f(x) + 3g(x)\bigr) = 2(2) + 3(4)\)
\(4 + 12\)
\(16\)
Answer: \(16\)
Graph of f
There is a vertical asymptote at \(x = 1\).
The left side goes up and the right side goes down, so the two-sided limit does not exist.
does not exist
Answer: does not exist
Graph of f
\(f(4) = 2\)
\(\lim_{x \to 4} f(x) = 2\)
\(\lim_{x \to 4} f(x) = f(4)\)
Yes, \(f\) is continuous at \(x = 4\).
Answer: yes
Graph of f
There is a vertical asymptote at \(x = 1\).
\(\lim_{x \to 1} f(x)\) does not exist, and \(f\) is not defined at 1.
No.
Answer: no
Graph of f
| \(x\) | 1.8 | 1.9 | 2 | 2.1 | 2.2 |
|---|---|---|---|---|---|
| \(g(x)\) | 0.4 | 0.7 | 1.0 | 1.3 | 1.6 |
From the table, \(g(x) \to 1\) as \(x \to 2\).
\(\lim_{x \to 2} f\bigl(g(x)\bigr) = \lim_{y \to 1} f(y)\)
\(f\) has a vertical asymptote at \(x = 1\), so that limit does not exist.
does not exist
Answer: does not exist
Graph of f
As \(x \to 2\), the middle piece passes through \((2, 3)\), so \(f(x) \to 3\).
\(\lim_{x \to 2} f\bigl(f(x)\bigr) = \lim_{y \to 3} f(y)\)
Near \(x = 3\) the graph is the middle piece, height 4 at \(x = 3\).
\(4\)
Answer: \(4\)
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