calculus-lessons/cheatsheet_limits.md

Here is the ultimate, step-by-step combat manual for tackling any limit problem in Calculus I. Bookmark this, print it out, or keep it on a second monitor while you do your homework. 

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)
**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 |
| :--- | :--- | :--- |
| **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). |
| $\frac{\text{Non-Zero}}{0}$ (e.g., $\frac{5}{0}$) | You hit a **Vertical Asymptote** (Infinite Discontinuity). | Move to **Step 3** (The Left/Right Test). |

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### Step 2: Rescue Tactics for $\frac{0}{0}$
If you got $\frac{0}{0}$, the limit *does* exist, but it is hiding in disguise. You have to change the way the equation looks so you can cancel out the broken part. Look at the equation and pick the right tool:

* **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 of the fraction by the exact same square root expression, but flip the middle sign (e.g., multiply by $\sqrt{x+4} + 2$). This forces the square roots to dissolve, allowing you to cancel.
* **Tool C: Common Denominators (For Fractions inside 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 big fraction. The broken part will reveal itself and cancel out.

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### Step 3: Handling $\frac{\text{Number}}{0}$ (Vertical Asymptotes)
If you get a normal number divided by zero, the graph is exploding to infinity or negative infinity. The overall limit **Does Not Exist (DNE)**, but your professor usually wants to know *which direction* it is exploding.

* **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 don't care about the exact number; 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 just **DNE**.

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### Step 4: The "Infinity" Shortcut (Limits as $x \to \infty$)
Sometimes a problem asks what happens as $x$ gets infinitely large: $\lim_{x \to \infty}$. Do not try to plug infinity in. Instead, simply 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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Would you like to grab a specific problem from your MyMathLab homework right now and use this cheat sheet to conquer it step-by-step?