From f12b7661b71cdbc32c143e91d7ad37267a24bb82 Mon Sep 17 00:00:00 2001 From: hugh Date: Sat, 28 Mar 2026 03:47:43 -0500 Subject: [PATCH] Delete cheatsheet_limits_2.md --- cheatsheet_limits_2.md | 51 ------------------------------------------ 1 file changed, 51 deletions(-) delete mode 100644 cheatsheet_limits_2.md diff --git a/cheatsheet_limits_2.md b/cheatsheet_limits_2.md deleted file mode 100644 index 469ccf2..0000000 --- a/cheatsheet_limits_2.md +++ /dev/null @@ -1,51 +0,0 @@ -# The Ultimate Limits Cheat Sheet - -### Step 1: The Golden Rule (Direct Substitution) -**Always plug the target $x$-value into the equation first.** The result will tell you exactly which path to take. - -| 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. | -| $\frac{0}{0}$ | You hit a **Hole** (Removable Discontinuity). | Move to **Step 2** (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). | - -### Step 2: Rescue Tactics for $\frac{0}{0}$ -If you get $\frac{0}{0}$, the limit *does* exist, but it is hiding in disguise. - -* **Tool A: Factoring (For Polynomials)** - * *When to use:* You see $x^2$ or $x^3$. - * *How:* Factor the top, factor the bottom. Cross out the matching groups. Plug your $x$-value into whatever is left. -* **Tool B: The Conjugate (For Square Roots)** - * *When to use:* You see a square root like $\sqrt{x+4} - 2$. - * *How:* Multiply the top and bottom by the exact same square root expression, but flip the middle sign (e.g., multiply by $\sqrt{x+4} + 2$). This forces the roots to dissolve, allowing you to cancel. -* **Tool C: Common Denominators (For Complex Fractions)** - * *When to use:* You see fractions stacked on top of other fractions. - * *How:* Combine the mini-fractions by finding a common denominator, then flip and multiply to simplify. The broken part will reveal itself. - -### Step 3: Handling $\frac{\text{Non-Zero}}{0}$ (Vertical Asymptotes) -If you get a normal number divided by zero, the graph is exploding. The overall limit **Does Not Exist (DNE)**, but you need to find the direction. - -* **The Tactic:** Pick a decimal slightly to the left (e.g., $1.99$) and slightly to the right (e.g., $2.01$) of your target $x$-value. -* **The Goal:** You only care about the **Sign**. - * If both sides equal a massive positive number, the answer is $\infty$. - * If both sides equal a massive negative number, the answer is $-\infty$. - * If one side is positive and the other is negative, the answer is **DNE**. - -### Step 4: The "Infinity" Shortcut ($\lim_{x \to \infty}$) -When a problem asks what happens as $x$ gets infinitely large, do not try to plug infinity in. Compare the **highest exponent on the top** to the **highest exponent on the bottom**. - -| The Scenario | The Shortcut Rule | Example | Answer | -| :--- | :--- | :--- | :--- | -| **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 $-\infty$. | $\lim_{x \to \infty} \frac{x^3}{5x + 2}$ | $\infty$ | -| **Perfect Tie** | The limit is the **ratio of the front numbers**. | $\lim_{x \to \infty} \frac{4x^2}{2x^2 + 1}$ | **$2$** (because $4/2 = 2$) | - ---- - -### When Do I Actually Use My Calculator? - -While Desmos is incredible for visualizing homework, you will strictly rely on your physical scientific calculator in a few specific scenarios, especially since your final exam will be proctored: - -1. **Executing "Step 1" (Direct Substitution):** When you are plugging in your initial $x$-value to see if you get $\frac{0}{0}$ or a normal number, let the calculator do the heavy lifting with the fractions and exponents so you don't make a simple arithmetic mistake. -2. **Executing "Step 3" (The Left/Right Test):** When you hit a vertical asymptote and need to plug in decimals like $1.99$ and $2.01$ to see if the equation explodes into positive or negative territory, your scientific calculator is essential. -3. **The "I'm Completely Stuck" Backup Plan:** If you get $\frac{0}{0}$ on a test, and you completely forget how to factor the equation, you can use your calculator to manually build a numerical table. Just plug in $1.9$, $1.99$, and $1.999$ into your calculator and see what number the answers are creeping toward. It is a brute-force method, but it works! \ No newline at end of file