Curriculum / Math / 11th Grade / Unit 9: Limits and Continuity / Lesson 7
Math
Unit 9
11th Grade
Lesson 7 of 9
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State and evaluate limits algebraically.
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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
LO2.1A, LO2.1B (approaching)
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Below is a piecewise function.
$${f(x)\left\{\begin{matrix}-x-2, \space \space -2\leq x <0 \\3x-2, \space \space \space 0\leq x <1 \\ x-3, \space \space \space 1\leq x \leq 4 \end{matrix}\right.}$$
Calculate the following:
$${\lim_{x\rightarrow 0}f(x)=}$$
$${\lim_{x\rightarrow 1}f(x)=}$$
How can you tell if the function is continuous without graphing?
Use $${f(x)={{x^2+6x+8}\over{x+2}}}$$ to evaluate:
a. $${\lim_{x\rightarrow -2} f(x)=}$$
b. $${\lim_{x\rightarrow 2} f(x)=}$$
c. $${f(-2)=}$$
d. $${f(2)=}$$
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Use $${f(x)={{x^2-7x+6}\over{x-6}}}$$ to evaluate:
a. $${\lim_{x\rightarrow6}f(x)=}$$
b. $${\lim_{x\rightarrow-1}f(x)=}$$
c. $${\lim_{x\rightarrow0}f(x)=}$$
d. $${f(6)=}$$
e. $${f(-1)=}$$
f. $${f(0)=}$$
Use $$g(x)=\left\{\begin{matrix} x+2 & x<-1 \\ x^2 & -1 \leq x <2\\ -2 x + 8 & 2 < x \leq 4 \end{matrix}\right.$$ to evaluate:
a. $${\lim_{x\rightarrow-1} g(x)=}$$
b. $${\lim_{x\rightarrow2}g(x)=}$$
c. $${g(2)=}$$
d. $${g(4)=}$$
e. $${g(-1)=}$$
f. $${\lim_{x\rightarrow-\infty}g(x)=}$$
Is this function $$g$$ continuous over the interval $${[0, 4]}$$? How do you know?
Lesson 6
Lesson 8
Topic A: Limits and Continuity
Graph, write, and evaluate linear piecewise functions.
Use interval and function notation to describe the behavior of piecewise functions.
Sketch a slope graph from a linear piecewise function.
Find limits, including left- and right-hand limits, on a function given graphically.
Define continuity of functions and determine whether a function is continuous on a particular domain.
Write and evaluate piecewise functions algebraically and graphically using parent functions.
Evaluate infinite limits and limits at infinity.
Sketch functions given limits and continuity requirements.
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