calculus-lessons/cheatsheet_limits_2.md

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!