Sketch functions given limits and continuity requirements.
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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 $${f(x)}=\begin{Bmatrix} x^2-5 & x\leq 3 \\ x+2 & x>3 \end{Bmatrix}$$
Write your answers in limit notation.
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.
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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.