Selecting Procedures for Determining Limits
Introduced| English | Meaning |
|---|---|
| flowchart/ˈfləʊtʃɑːt/ | A diagram showing the sequence of steps and decisions in a process or algorithm. |
| procedure/prəˈsiːdʒə/ | A quick flow: substitute first. |
A limit is a "which tool?" puzzle
- You now own several ways to find a limit — the skill is picking the right one fast.
- Rushing into heavy algebra when substitution would work wastes time; guessing when it gives $\tfrac00$ wastes marks.
- Think of every limit as a short decision: look, try, then choose.
- This lesson is the flowchart 流程图 that ties the earlier tools together.
The decision flow
- Step 1 — try direct substitution. Plug $c$ in.
- If you get a real number, that is the limit. Done.
- If you get $\dfrac{0}{0}$ (indeterminate), go to Step 2.
- If you get $\dfrac{\text{nonzero}}{0}$, the size blows up — check for a vertical asymptote (an infinite limit or DNE), not algebra.
Probe a rational function
y = ax² + bx + c
Change the shape and ask whether plugging in gives a number, a hole ($\tfrac00$), or a blow-up ($\tfrac{n}{0}$) — that triage picks your tool.
Direct substitution into $\lim_{x\to 4}\dfrac{x-4}{x^2-16}$ gives $\tfrac00$. What next?
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Polynomial over polynomial, both zero at $4$ → factor: $\frac{x-4}{(x-4)(x+4)}=\frac{1}{x+4}\to\frac18$.
The first procedure to attempt on almost any limit is direct ____.
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It is fastest when it works and tells you which branch to take when it does not.
Step 2 — match the form to a technique
- Polynomial over polynomial, both zero at $c$ → factor and cancel.
- A square root causing the zero → rationalize with the conjugate.
- A difference of fractions → combine over a common denominator, then cancel.
- Special famous forms (like $\frac{\sin x}{x}$) → recognize the known limit or the Squeeze Theorem.
Substitution into $\lim_{x\to 3}\dfrac{x+1}{x-3}$ gives $\dfrac{4}{0}$. This signals...
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Nonzero over zero blows up; factoring cannot help. The two-sided limit is DNE (one side $+\infty$, other $-\infty$).
Select all correct method-to-form matches.
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The first three match. Nonzero over zero is an asymptote, not something factoring resolves.
Evaluate $\displaystyle\lim_{x\to 5}\dfrac{x^2-25}{x-5}$.
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$\tfrac00$ → factor: $x+5 \to 10$.
Step 3 — justify and sanity-check
- State why your method is valid: "substitution works because the function is continuous here," or "cancelling is valid for $x\neq c$."
- Check the answer is consistent — does a quick table or the graph's shape agree?
- A good solution names its procedure 步骤 and confirms the result, not just a bare number.
- Efficiency comes from choosing well the first time, not from trying everything.
$\tfrac00$ and $\tfrac{5}{0}$ should be handled the same way.
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$\tfrac00$ is indeterminate (simplify); $\tfrac{5}{0}$ blows up (asymptote). Different situations, different responses.
Do not confuse the two "over zero" outcomes. $\dfrac00$ is indeterminate → simplify with algebra. But $\dfrac{5}{0}$ (nonzero over zero) is not indeterminate → the magnitude grows without bound, so you are looking at an infinite limit / vertical asymptote, and factoring will not help.
Choose a method for each, then evaluate:
- $\displaystyle\lim_{x\to 2}(x^2+1)$: substitution → $5$. (Continuous — no tricks.)
- $\displaystyle\lim_{x\to 5}\dfrac{x^2-25}{x-5}$: substitution gives $\tfrac00$ → factor → $x+5 \to 10$.
- $\displaystyle\lim_{x\to 3}\dfrac{1}{x-3}$: substitution gives $\tfrac{1}{0}$ → not algebra; the one-sided limits are $\pm\infty$, so the two-sided limit is DNE.
Selecting a procedure is a quick flow: substitute first. A real number is the answer; $\tfrac00$ means simplify (factor / rationalize / combine, or a known limit); $\tfrac{\text{nonzero}}{0}$ means an infinite limit or asymptote, not algebra. Then justify the choice and sanity-check the result.
Write from choosing a tool
- One limit substitutes to 7.
- Another substitutes to a non-zero number over 0.
Which result is the answer, and what does the other form mean?
The form tells you the next tool.
Direct substitution gives 7. What should you do?
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A real number from substitution is the limit.