Limits and Continuity
Benjamin Wright
10 min read
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Study Guide Overview
This study guide covers limits and continuity in AP Calculus AB/BC. Key topics include: average and instantaneous rates of change, defining and estimating limits (from graphs and tables), algebraic properties and manipulations of limits, the Squeeze Theorem, types of discontinuities, defining continuity at a point and over an interval, removing discontinuities, asymptotes, and the Intermediate Value Theorem (IVT).
#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
- Average Rate of Change (AROC): The slope of the secant line between two points on a function. It's like finding the average speed over a time interval.
- Formula:
- Instantaneous Rate of Change (IROC): The slope of the tangent line at a single point on a function. It's like finding the speed at a specific moment. We'll get to this using limits!
- Formula:
#Image courtesy of Medium.
- AROC is undefined when the denominator is zero (division by zero is a big no-no!).
#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
- Read this as: "The limit of f(x) as x approaches 'a' is L."
- Think of it like this: As x gets super close to 'a', the function f(x) gets super close to L.
#1.3 Estimating Limits from Graphs
#One-Sided vs. Two-Sided Limits
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One-Sided Limit: The limit as x approaches a value from either the left or the right.
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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.
- If the function appro...
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