# Limits and Continuity

## Objective

Evaluate infinite limits and limits at infinity.

## Criteria for Success

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

## Anchor Problems

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### Problem 1

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?

### Problem 2

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

## Problem Set

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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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### Problem 1

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

### Problem 2

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?