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Update cheatsheet_limits.md

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hugh
2026-03-27 18:23:28 -05:00
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@@ -3,13 +3,13 @@ Here is the ultimate, step-by-step combat manual for tackling any limit problem
When you are given a problem asking for $\lim_{x \to a} f(x)$, follow this exact sequence: When you are given a problem asking for $\lim_{x \to a} f(x)$, follow this exact sequence:
### Step 1: The Golden Rule (Direct Substitution) ### Step 1: The Golden Rule (Direct Substitution)
**Always, always, always plug the target $x$-value into the equation first.** Do not start doing crazy math until you see what happens when you just plug the number in. The result will tell you exactly which of the three paths you need to take. **Always, always, always plug the target x-value into the equation first.** Do not start doing crazy math until you see what happens when you just plug the number in. The result will tell you exactly which of the three paths you need to take.
| What you get | What it means | What you do | | What you get | What it means | What you do |
| :--- | :--- | :--- | | :--- | :--- | :--- |
| **A normal number** (e.g., $4$, $-10$, $0$) | The graph is perfectly continuous here. | **You are done.** That number is your answer. | | **A normal number** (e.g., $4$, $-10$, $0$) | The graph is perfectly continuous here. | **You are done.** That number is your answer. |
| **$\frac{0}{0}$** | You hit a **Hole** (Removable Discontinuity). | Move to **Step 2** (The Rescue Tactics). | | $\frac{0}{0}$ | You hit a **Hole** (Removable Discontinuity). | Move to **Step 2** (The Rescue Tactics). |
| **$\frac{\text{Non-Zero}}{0}$** (e.g., $\frac{5}{0}$) | You hit a **Vertical Asymptote** (Infinite Discontinuity). | Move to **Step 3** (The Left/Right Test). | | $\frac{\text{Non-Zero}}{0}$ (e.g., $\frac{5}{0}$) | You hit a **Vertical Asymptote** (Infinite Discontinuity). | Move to **Step 3** (The Left/Right Test). |
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@@ -44,9 +44,9 @@ Sometimes a problem asks what happens as $x$ gets infinitely large: $\lim_{x \to
| The Scenario | The Shortcut Rule | Example | Answer | | The Scenario | The Shortcut Rule | Example | Answer |
| :--- | :--- | :--- | :--- | | :--- | :--- | :--- | :--- |
| **Bottom Heavy** | The limit is always **$0$**. | $\lim_{x \to \infty} \frac{3x}{x^2 + 1}$ | **$0$** | | **Bottom Heavy** | The limit is always $0$. | $\lim_{x \to \infty} \frac{3x}{x^2 + 1}$ | $0$ |
| **Top Heavy** | The limit goes to **$\infty$** or **$-\infty$**. | $\lim_{x \to \infty} \frac{x^3}{5x + 2}$ | **$\infty$** | | **Top Heavy** | The limit goes to $\infty$ or $-\infty$. | $\lim_{x \to \infty} \frac{x^3}{5x + 2}$ | $\infty$ |
| **Perfect Tie** | The limit is the **ratio of the front numbers**. | $\lim_{x \to \infty} \frac{4x^2}{2x^2 + 1}$ | **$2$** (because $4/2 = 2$) | | **Perfect Tie** | The limit is the **ratio of the front numbers**. | $\lim_{x \to \infty} \frac{4x^2}{2x^2 + 1}$ | $2$ (because $4/2 = 2$) |
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