Sketch functions given limits and continuity requirements.

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- Use specifications about limits to sketch functions.
- Identify and use continuity requirements to sketch functions.
- Use limit notation appropriately for right-hand, left-hand, and actual limits.

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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

Note: This lesson will also serve as the review for this unit.

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For the function $${f(x)}$$, given below, determine the following:

Let $${f(x)}=\begin{Bmatrix} x^2-5 & x\leq 3 \\ x+2 & x>3 \end{Bmatrix}$$

- The left-hand limit at $${{{{x=3}}}}$$
- The right-hand limit at $${{{{x=3}}}}$$
- The limit at $${{{{x=3}}}}$$
- The value of the function at $${{{{x=3}}}}$$

Write your answers in limit notation.

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Draw a picture of a function with the following properties:

$${\lim_{x\rightarrow a^+}f(x)\neq \lim_{x\rightarrow a^-}f(x)}$$

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

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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.

- Use this problem set as an opportunity review all concepts from the unit.
- Include error analysis by giving a list of requirements and a graph with errors. Ask students to adjust the requirements to meet the graph or adjust the graph to meet the requirements.
- Give requirements that indicate different types of point discontinuities and jump discontinuities.

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Draw the picture of a function where $${f(c)=\lim_{x\rightarrow c^+}f(x)}$$, but the limit at $${x=c}$$ does not exist.