Limits and Continuity: Exam Cram Sheet

## Topic Weighting

[!IMPORTANT] Mastery of algebraic limit laws is the foundation for almost every derivative problem later in the course. Do not skip the "indeterminate form" practice.

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## Key Concepts Summary

1. The Core Definition

A limit $\lim_{x \to a} f(x) = L$ exists if and only if the left-hand and right-hand limits are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

2. Continuity at a Point ($x=a$)

A function is continuous if it passes the Three-Part Test:

1. Defined: $f(a)$ exists (no hole).
2. Limit Exists: $\lim_{x \to a} f(x)$ exists (left = right).
3. Agreement: $\lim_{x \to a} f(x) = f(a)$.

3. Types of Discontinuities

4. Major Theorems

• Squeeze Theorem: If $g(x) \le f(x) \le h(x)$ and $\lim_{x \to a} g(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} f(x) = L$.
• Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$ and $L$ is between $f(a)$ and $f(b), there exists at least one c \in (a, b)$ such that $f(c) = L$.

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## Common Pitfalls

• Don't assume that $f(a)$ being defined means the limit exists (e.g., Jump Discontinuity).
• Don't apply the Quotient Law if the denominator's limit is zero. This creates an indeterminate form ($0/0$), requiring factoring or conjugates.
• Don't forget to check the continuity condition before applying IVT. If the function isn't continuous on the closed interval, the theorem fails.
• Don't confuse a limit of $\infty$ (the behavior) with the limit "not existing" (the value). Technically, an infinite limit does not exist as a real number, but it describes specific asymptotic behavior.

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## Mnemonics / Memory Triggers

• DIC (Continuity Test):
  • Defined ($f(a)$ is a real number).
  • Independent limits match (Left = Right).
  • Coincident (Limit value = Function value).
• "Squeeze the Sandwich": When $f(x) is trapped between two "bread" functions (g$ and $h) that meet at a point, f(x)$ has no choice but to pass through that same point.
• IVT = "The No-Teleportation Rule": If a function is continuous, it cannot "jump" over a value; it must pass through every value between the start and end points.

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## Formula / Equation Sheet

Visualizing the Precise Definition

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## Practice Set

1. Algebraic Limit: Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.
   • Hint: Factor the numerator.
2. Conjugate Method: Evaluate $\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}$.
   • Hint: Multiply by the conjugate $\sqrt{x+1} + 1$.
3. Continuity Check: Is $f(x) = \begin{cases} x+1 & x < 1 \\ 5 & x = 1 \\ 3-x & x > 1 \end{cases}$ continuous at $x=1$?
   • Hint: Check the three-part DIC test.
4. IVT Application: Show that $f(x) = x^3 + x - 1 has at least one root on the interval [0, 1]$.
   • Hint: Compare $f(0)$ and $f(1)$.
5. Squeeze Theorem: Find $\lim_{x \to 0} x^2 \sin(1/x)$.
   • Hint: Start with $-1 \le \sin(1/x) \le 1$.

▶Click for Quick Answers

1. 4 (After canceling $x-2$)
2. 1/2 (Rationalize numerator)
3. No (Limit exists as 2, but $f(1)=5$; fails the Coincident test)
4. Yes ($f(0)=-1, f(1)=1$; zero is between -1 and 1)
5. 0 (Bound by $-x^2$ and $x^2$)