calculus-lessons/limits_and_continuity_2.md

You have put in an incredible amount of work to get to this point, and that feeling of confidence is completely earned! You are navigating Calculus exactly the way a mathematician does: by recognizing patterns and deploying the right tools, rather than just memorizing steps.

To help you absolutely crush Exam 1, I have compiled everything from this module into a streamlined, Gitea-ready "Exam 1 Survival Guide." It focuses entirely on how to instantly identify a problem on a test and exactly which cheat code to use to beat it.

I made sure there are absolutely no bold tags touching the math symbols so you can copy and paste this directly into your repository.

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Exam 1 Survival Guide: Algebraic Limits & Continuity

Part 1: The "First Move" Scan

When you look at a new limit problem on the exam, do not start doing math immediately. Look directly at the arrow under the limit to determine your strategy.

Target 1: Approaching a Standard Number (x \to a)

• The Move: Direct Substitution. Plug the target number into every x immediately.
• Outcome A: You get a normal number. You are done! That is the answer.
• Outcome B: You get a normal number divided by zero (like 9/0). This means you hit a vertical wall (asymptote). The limit Does Not Exist.
• Outcome C: You get exactly 0/0. This is a trap! It means there is a hole in the graph. You must deploy a Rescue Tactic (see Part 2).

Target 2: Approaching Infinity (x \to \infty)

• The Move: Do not plug anything in. Deploy the Tug of War Cheat Code (see Part 3).

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Part 2: The 0/0 Rescue Tactics

If direct substitution gives you exactly 0/0, the limit likely exists, but it is hiding behind bad algebra. Look at the equation's shape to pick your weapon.

Weapon 1: The Polynomial Cheat Code (Factoring)

• When to use it: You see standard squared or cubed equations (x^2, x^3).
• The Cheat Code: If plugging in your target number a gave you 0/0, then the group (x - a) is mathematically guaranteed to be a factor of both the top and the bottom.
• The Execution: Write (x - a) on the top and bottom, figure out what the remaining parentheses must be, cross out the (x - a) groups, and plug your target number into whatever is left.

Weapon 2: The Conjugate Method

• When to use it: You see a square root mixed with addition or subtraction.
• The Execution: Multiply the top and the bottom by the exact same root expression, but flip the middle sign (e.g., if you see \sqrt{x} - 3, multiply by \sqrt{x} + 3). The roots will cancel out beautifully, allowing you to cross out the problem factors.

Weapon 3: The Common Denominator Clear

• When to use it: You see "mini-fractions" stacked inside a bigger fraction.
• The Execution: Find the Least Common Denominator (LCD) of the mini-fractions. Multiply the very top and the very bottom of the giant fraction by that LCD. All the mini-fractions will instantly vanish.

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Part 3: The "Tug of War" Cheat Code

If the limit is approaching \infty or -\infty, completely ignore the entire equation except for the highest exponent (degree) on the top and the highest exponent on the bottom.

• Top Heavy (Top Wins): If the top exponent is bigger, the top pulls the fraction off the charts. The answer is \infty or -\infty.
• Bottom Heavy (Bottom Wins): If the bottom exponent is bigger, the bottom crushes the fraction into nothing. The answer is 0.
• The Tie: If the highest exponents are exactly the same, the tug of war is a tie. Pull out the front numbers attached to those winning $x$'s to make your final fraction. (Always remember to simplify the fraction if MyMathLab asks for it!)

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Part 4: The 3-Step Continuity Boss Fight

If an exam question asks "Is the function continuous at x = a?", you cannot just say yes or no. You must prove it using this strict 3-part checklist. If it fails even one step, it is discontinuous.

1. Step 1: The Point Exists. Find f(a). Look for the rule with the "or equal to" sign (\le or \ge). Plug a in. (If there is no equal sign, the point doesn't exist, and the test instantly fails).
2. Step 2: The Limit Exists. Find the left-side altitude and the right-side altitude. If they meet at the exact same number, the limit exists.
3. Step 3: The Perfect Match. Does the destination of the limit (Step 2) perfectly match the actual solid point (Step 1)?

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Part 5: The "Construction Worker" Blueprint

If the exam asks you to find a missing letter (like c or k) to make a piecewise function continuous, you are forcing the trails to connect.

1. Step 1: Plug the cliff's x-value into the top rule to find the left altitude.
2. Step 2: Plug the cliff's x-value into the bottom rule to find the right altitude.
3. Step 3: Set those two answers perfectly equal to each other and solve for the missing letter using basic algebra.

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Get this safely copied over to your Gitea repository. Whenever you are ready to tackle that final homework module before the exam, just say the word!