Limits and Continuity

Lesson 4

Objective

Find limits, including left- and right-hand limits, on a function given graphically. 

Criteria for Success

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  1. Describe that a limit is the $${y-}$$value that a function gets arbitrarily close to as the function gets closer to a named $${x-}$$value. 
  2. Use limit notation to describe limits from the left-hand, right-hand, and overall limit. 
  3. Read the notation $${\lim_{x\rightarrow1^+}f(x)}$$ as “The limit of $$f$$ of $$x$$ as $$x$$ approaches $$1$$ from the right side.”
  4. Read the notation $$\lim_{x\rightarrow1^-}f(x)$$ as “The limit of $$f$$ of $$x$$ as $$x$$ approaches $$1$$ from the left side.”
  5. Read the notation $$\lim_{x\rightarrow1}f(x)$$ as “The limit of $$f$$ of $$x$$ as $$x$$ approaches $$1$$.”
  6. Describe that if the left-hand and right-hand limits are not equal, the limit of the function at that point does not exist (DNE).

Tips for Teachers

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This lesson is aligned to the Learning Objectives and Essential Knowledge described in the College Board's AP Calculus AB and AP Calculus BC Course and Exam Description:

LO1.1A(b): EK1.1A1, EK1.1A2, EK1.1A3

LO1.1B: EK1.1B1

LO1.2A: EK1.2A1

Anchor Problems

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Problem 1

Below is a linear piecewise function.

If you travel along the graph from point $$A$$ to point $$B$$, what $${{y-}}$$value do you get closer to as you get closer to $${{x=7}}$$

If you travel along the graph from point $$C$$ to point $$B$$, what $${{y-}}$$value do you get closer to as you get closer to $${{x=7}}$$?

Guiding Questions

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Problem 2

Find the following limits:

$${\lim_{x\rightarrow4^+}f(x)=}$$

$${\lim_{x\rightarrow4^-}f(x)=}$$

$${\lim_{x\rightarrow4}f(x)=}$$

 

 

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Using piecewise functions where there is a point discontinuity, a jump, and continuous piecewise functions, identify limits. In this lesson, we will only identify limits graphically, not algebraically.
  • Start to include a few non-linear piecewise functions for this lesson. Stick to basic sine, cosine, quadratic cubic, and square root functions. 
  • Include the problem type where students are given multiple choice and then asked to change the other answer choices to make true statements. 

Target Task

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Based on the graph below, which of the following is falase? Circle ALL that apply. Then, change the false statements into true statements.

 

A.  $${\lim_{x\rightarrow 1^+}h(x)\neq \lim_{x\rightarrow 1^-}h(x)}$$

B.  $${\lim_{x\rightarrow 1}h(x)\space \mathrm{exists}}$$

C.  $${\lim_{x\rightarrow 1^-}h(x)\neq h(1)}$$

D.  $${\lim_{x\rightarrow 1^+}h(x)\neq h(1)}$$