# 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 $4/2 = 2$) |

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### 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!