Evaluate infinite limits and limits at infinity.

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- Identify the end behavior of a function algebraically and graphically.
- Use the end behavior of a function to find limits at infinity.
- Evaluate left- and right-hand limits at vertical asymptotes.

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Carla makes $${21}$$ of her first $${30}$$ free throws of the basketball season and then goes on a streak making every shot after that. Her free throw percentage is modeled by the function $$P(m)={{m+{21}}\over{m+{30}}}$$. As she takes more shots, what does her free throw percentage get closer to?

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Use $${f(x)={{x+2}\over{x^2+4x+4}}}$$ to evaluate:

a. $${\lim_{x\rightarrow -2}f(x)=}$$ |
b. $${\lim_{x\rightarrow 2}f(x)=}$$ |

c. $${f(-2)=}$$ |
d. $${f(2)=}$$ |

e. $${\lim_{x\rightarrow \infty}f(x)=}$$ |
f. $${\lim_{x\rightarrow -\infty}f(x)=}$$ |

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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.

- Include problems finding limits with only a graph, with only an algebraic function, and both
- Include problems with finding limits at infinity that tend toward negative or positive infinity, as well as ones that have horizontal asymptotes
- While this is not included in the anchor problems, it could be useful to find limits at infinity of functions like $${\mathrm{ln}x}$$, $${\mathrm{sin}x}$$, and $${{\mathrm{sin}x}}\over x$$.
- Include functions like $${1\over x}$$ that have different limits on different sides of a vertical asymptote, and functions like $${1\over x}^2$$ that have the same limit on each side
- Include the following problem:

Use $${f(x)={{(x+10)(x-2)}\over{(x+4)(x-3)}}}$$ to evaluate:

a. $${\lim_{x\rightarrow -4}f(x)=}$$ |
b. $${\lim_{x\rightarrow 3}f(x)=}$$ |

c. $${\lim_{x\rightarrow 2}f(x)=}$$ |
d. $${f(-4)=}$$ |

e. $${f(3)=}$$ |
f. $${f(2)=}$$ |

g. $${\lim_{x\rightarrow \infty}f(x)=}$$ |
h. $${\lim_{x\rightarrow -\infty}f(x)=}$$ |

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Use $${f(x)={{x^2-7x+12}\over{x-6}}}$$ to evaluate:

a. $${\lim_{x\rightarrow6}f(x)=}$$ |
b. $${f(6)=}$$ |

c. $${\lim_{x\rightarrow\infty}f(x)=}$$ |
d. $${\lim_{x\rightarrow-\infty}f(x)=}$$ |

Use $${g(x)=\left\{\begin{matrix} x+2, & x<-1\\ (x+1)^2+1, & -1\leq x <2 \\ -{1\over x}, & 2<x\leq4 \end{matrix}\right.}$$ to evaluate:

a. $${\lim_{x\rightarrow-\infty}g(x)=}$$ |
b. $${\lim_{x\rightarrow\infty}g(x)=}$$ |

Is this function $$g$$ continuous over the interval $${[0, 4]}$$? How do you know?