#AP Calculus AB/BC: Limits and Continuity - Your Night-Before Guide 🚀

Hey there, future calculus conqueror! This guide is your go-to resource for a final review of limits and continuity. Let's make sure you're feeling confident and ready to ace that exam! Remember, this unit is about 10-12% of the AP Calculus AB exam and 4-7% of the AP Calculus BC exam, so let's get it down!

#Introduction to Calculus

#The Big Question: Can Change Happen Instantly?

Calculus is all about change and motion. We're diving into whether change can occur at a single moment. Think of an arrow moving across a screen. It seems to jump from one spot to the next, but is it truly instantaneous? This leads us to the concept of the limit.

#1.1 Introducing Calculus: Average vs. Instantaneous Rate of Change

#Image courtesy of Medium.

#1.2 Defining Limits and Using Limit Notation

#What is a Limit?

A limit is the value a function approaches as the input (x-value) gets closer to a certain point. It helps us understand what happens at a specific point, even if the function isn't defined there. 💡

#Limit Notation

$$\lim\_{x \to a} f(x) = L$$

#1.3 Estimating Limits from Graphs

#One-Sided vs. Two-Sided Limits

• One-Sided Limit: The limit as x approaches a value from either the left or the right.

• Two-Sided Limit: The limit as x approaches a value from both the left and the right. For a two-sided limit to exist, both one-sided limits must be equal.

#1.4 Estimating Limits from Tables

#Approaching from Both Sides

• Look at the function's values as x gets closer to the target value from both the left and the right.
• If the function approaches the same value from both sides, the limit exists. If not, the limit does not exist.

#1.5 Using Algebraic Properties of Limits

#Limit Laws

These properties help us break down complex limits into simpler ones. They’re pretty intuitive, but it’s good to be familiar with them.

#1.6 Using Algebraic Manipulations

#Four Key Techniques

1. Substitution: Plug in the value. If it works, you're done! (But be careful of division by zero!)

2. Factoring: Factor the numerator and/or denominator to cancel out common factors.

3. Common Denominator: Combine fractions by finding a common denominator.

4. Conjugate Multiplication: Multiply by the conjugate to eliminate square roots in the numerator or denominator.

#1.8 Squeeze Theorem (Sandwich Theorem)

#The Pinching Principle

If a function is "sandwiched" between two other functions that approach the same limit, then the function in the middle also approaches that limit.

#1.10 Exploring Types of Discontinuities

#Discontinuity Types

• Jump Discontinuity: The function "jumps" from one value to another.
• Removable Discontinuity: A "hole" in the graph that can be "filled" by redefining the function.
• Infinite Discontinuity: A vertical asymptote where the function approaches infinity or negative infinity.

#Image courtesy of Matlab and Maths Tutorials.

#1.11 Defining Continuity at a Point

#The Three Requirements for Continuity

For a function to be continuous at a point x=a, three conditions must be met:

1. f(a) is defined (there's a y-value at x=a).

2. The limit of f(x) as x approaches a exists.

3. The limit of f(x) as x approaches a is equal to f(a).

#1.12 Confirming Continuity Over an Interval

#Continuity Over an Interval

A function is continuous over an interval if it meets the three continuity conditions at every point in that interval.

#Common Continuous Functions

• Polynomials: Always continuous everywhere.

• Rational Functions: Continuous everywhere except where the denominator is zero.

• Trigonometric Functions: Continuous over their domains.

• Exponential Functions: Continuous everywhere.

#1.13 Removing Discontinuities

#Redefining the Function

If a limit exists at a point of discontinuity, you can redefine the function at that point to make it continuous. This usually involves creating a piecewise function.

#1.14 & 1.15 Asymptotes

#Vertical Asymptotes

• Found where the denominator of a rational function equals zero.

#Horizontal Asymptotes

• Compare the degrees of the numerator and denominator:
  • Degrees are equal: Horizontal asymptote at the ratio of leading coefficients.
  • Numerator degree is larger: Slant asymptote.
  • Denominator degree is larger: Horizontal asymptote at y = 0. ## 1.16 Working with the Intermediate Value Theorem (IVT)

#The IVT Explained

If a function is continuous on a closed interval [a, b], it must take on every value between f(a) and f(b) at least once within that interval. It's like a continuous path that can't skip any values!

#Image courtesy of Quizlet.

#Final Exam Focus

#High-Priority Topics

• Limits: Evaluating limits graphically, numerically, and algebraically. Special attention to limits involving infinity.
• Continuity: Understanding the three conditions for continuity at a point and over an interval. Identifying and classifying discontinuities.
• IVT: Applying the Intermediate Value Theorem to prove the existence of solutions.

#Common Question Types

• Multiple Choice: Evaluating limits, identifying discontinuities, applying IVT.
• Free Response: Proving continuity, finding asymptotes, using the squeeze theorem, applying IVT in context.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. Move on and come back if you have time.
• Common Pitfalls: Double-check for division by zero, remember to check one-sided limits, and ensure continuity before applying IVT.
• Strategies: Show all work, even if you think it's obvious. Partial credit is your friend!

#Practice Questions

Practice Question

#Multiple Choice

1. What is the value of $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$? (A) 0 (B) 3 (C) 6 (D) Does not exist

2. For what value of k is the following function continuous at x = 2?

   $$f(x) = \begin{cases} x^2 + k, & x \leq 2 \\ 3x + 1, & x > 2 \end{cases}$$

   (A) -1 (B) 0 (C) 1 (D) 2

3. Given the function $f(x) = \frac{1}{x-2}$, which of the following statements is true? (A) f(x) has a removable discontinuity at x=2 (B) f(x) has a jump discontinuity at x=2 (C) f(x) has an infinite discontinuity at x=2 (D) f(x) is continuous for all real numbers

#Free Response Question

Consider the function:

$$f(x) = \begin{cases} \frac{x^2-4}{x-2}, & x < 2 \\ 5, & x = 2 \\ x + 2, & x > 2 \end{cases}$$

(a) Find $\lim_{x \to 2^-} f(x)$. (b) Find $\lim_{x \to 2^+} f(x)$. (c) Is f(x) continuous at x = 2? Justify your answer. (d) If f(x) is discontinuous at x=2, what type of discontinuity is it?

#Scoring Rubric:

(a) 2 points

• 1 point for factoring the expression
• 1 point for correct limit value

(b) 2 points

• 1 point for correct substitution
• 1 point for correct limit value

(c) 3 points

• 1 point for stating that the function is discontinuous
• 1 point for referencing that the limit does not equal the function value at x=2
• 1 point for correct justification

(d) 1 point

• 1 point for correctly identifying the type of discontinuity

#Solutions:

Multiple Choice

1. (C) 6
2. (C) 1
3. (C) f(x) has an infinite discontinuity at x=2

Free Response (a) $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} \frac{x^2-4}{x-2} = \lim_{x \to 2^-} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2^-} (x+2) = 4$ (b) $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x+2) = 2+2 = 4$ (c) No, f(x) is not continuous at x=2 because $\lim_{x \to 2} f(x) = 4$ which does not equal $f(2) = 5$ (d) f(x) has a removable discontinuity at x=2

You've got this! Go get that 5! 💪