AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/1/1-9)
Unit 1 · Topic 1.9
1.9 Connecting Multiple Representations of Limits
A limit can be shown with a graph, a table, a formula or a sentence. You should be able to read the limit from any of these, translate between them, and combine information from different ones in the same problem.
Key terms
- graphical
- numerical
- analytical
- verbal
- representation
Four representations
Each representation shows the same limit in a different way, and each has its own pitfalls:
| Representation | How you find the limit | Watch out for |
|---|---|---|
| Graphical (graph) | Trace the curve from each side toward x = c | Holes and dots, scale |
| Numerical (table) | Watch outputs as inputs approach c from each side | Too few points; it's an estimate |
| Analytical (formula) | Substitute, then rewrite if 0/0 | Piecewise breaks, zero denominators |
| Verbal (words) | Translate the sentence into limit notation | One-sided vs. two-sided wording |
Translating words into limits
“As x gets close to 2 from the left, f(x) gets close to 5” is lim (x→2⁻) f(x) = 5. “As t increases without bound, the population approaches 800” is lim (t→∞) P(t) = 800. Practice reading both directions: from notation to a sentence, and back.
Checking that representations agree
Sometimes a question gives a limit in one form and asks which other representation matches it. For example, if lim (x→3) f(x) = 2 but f(3) = 5, a matching graph has a hole at (3, 2) and a separate dot at (3, 5), and a matching table has outputs near 3 that close in on 2 from both sides.
A table can be consistent with a limit without proving it. A graph with a jump at c can't match a two-sided limit statement at c. A formula has the final say: when you have one, use algebra to confirm what the graph or table suggests.
Mixed problems
The exam likes to give f as a graph and g as a formula or table, then ask about f + g, f·g or f(g(x)). Find each limit with its own representation, then combine using the properties from 1.5.
Limits of composite functions in detail
For lim (x→c) f(g(x)), first find what g(x) approaches, say L. If f is continuous at L, the answer is f(L).
If f has a break at L, you also need to know which side g(x) approaches L from. If g(x) comes in from above L, use lim (u→L⁺) f(u). If it comes from below, use the left-hand limit. If g(x) equals L exactly for x near c, then f(g(x)) is just the constant f(L) there.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Combining a graph and a table
From the graph of f, lim (x→2) f(x) = 5. A table of g shows g(1.9) = 1.12, g(1.99) = 1.012, g(2.01) = 0.988 and g(2.1) = 0.88. Estimate lim (x→2) [f(x) + 3g(x)].
Show the solutionHide the solution
- Step 1: From the table, g(x) approaches 1 from both sides, so lim (x→2) g(x) appears to be 1.
- Step 2: Use the sum and constant multiple properties: lim [f(x) + 3g(x)] = 5 + 3(1).
Answer: The limit is approximately 8.
- Example 2
Trap: a composite limit at a jump
Let g(x) = 3 + (x − 1)², and let f(u) = 2 for u < 3 and f(u) = u + 1 for u ≥ 3. Find lim (x→1) f(g(x)). Note that lim (u→3) f(u) does not exist.
Show the solutionHide the solution
- Step 1: Inner limit: as x→1, g(x) → 3 + 0 = 3.
- Step 2: f has a jump at 3 (left-hand limit 2, right-hand limit 4), so you need to know how g(x) approaches 3.
- Step 3: Since (x − 1)² ≥ 0, g(x) ≥ 3 for every x. So g(x) approaches 3 from above (or equals 3 at x = 1, which a limit ignores).
- Step 4: Use the right-hand behavior of f: lim (u→3⁺) f(u) = 3 + 1 = 4.
Answer: lim (x→1) f(g(x)) = 4, even though lim (u→3) f(u) does not exist.
Common mistakes
- Assuming lim f(g(x)) doesn't exist just because f has a jump at the inner limit. Check which side the inner function comes from.
- Reading the wrong function's graph or table in a mixed problem. Label which representation belongs to f and which to g.
- Translating “approaches from the right” as x→c⁻. Right means larger x-values, so the superscript is +.
On the exam
- Multiple-choice questions often pair a graph with a formula or table and ask for a limit of a combination or composition.
- Verbal statements of limits show up later too, for example in describing end behavior of a model in context.
Connected topics
Videos
Check yourself
4 questions on 1.9 Connecting Multiple Representations of Limits. Pick an answer to see if you got it, and why.
Let f(x) = (x² − 4)/(x − 2) for x ≠ 2, and let f(2) = 1. Which of the following statements are true? I. lim (x→2) f(x) = 4 II. f is continuous at x = 2. III. The graph of f is the line y = x + 2 with an open circle at (2, 4), plus the point (2, 1).
A function f satisfies lim (x→2) f(x) = 5 and f(2) = 1. Each choice lists values of f(x) for x = 1.9, 1.99, 2, 2.01 and 2.1, in that order. Which list is consistent with this information?
A function h has a removable discontinuity at x = 3, and lim (x→3) h(x) = −2. Which of the following could be h(x)?
The function g is defined on the closed interval [−4, 4]. Its graph is described below.
On [−4, −1), the graph is a line segment from the point (−4, −2) to an open circle at (−1, 1).
The point (−1, 3) is on the graph, so g(−1) = 3.
On (−1, 2), the graph is a line segment from an open circle at (−1, 1) to an open circle at (2, −2).
On [2, 4], the graph is a line segment from the closed point (2, 0) to the point (4, 4).
Described graph of g
What is lim (x→−1) g(g(x)) ?
0 of 4 answered