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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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:
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a. $${\lim_{x\rightarrow -2} f(x)=}$$ |
b. $${\lim_{x\rightarrow 2} f(x)=}$$ |
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c. $${f(-2)=}$$ |
d. $${f(2)=}$$ |
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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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Use $${f(x)={{x^2-7x+6}\over{x-6}}}$$ to evaluate:
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a. $${\lim_{x\rightarrow6}f(x)=}$$ |
b. $${\lim_{x\rightarrow-1}f(x)=}$$ |
c. $${\lim_{x\rightarrow0}f(x)=}$$ |
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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:
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a. $${\lim_{x\rightarrow-1} g(x)=}$$ |
b. $${\lim_{x\rightarrow2}g(x)=}$$ |
c. $${g(2)=}$$ |
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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?