Limits and Continuity

Lesson 8

Math

Unit 9

11th Grade

Lesson 8 of 9

Objective


Evaluate infinite limits and limits at infinity.

Criteria for Success


  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.
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Anchor Problems

25-30 minutes


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?

Guiding Questions

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

Guiding Questions

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Target Task

5-10 minutes


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?

Next

Sketch functions given limits and continuity requirements.

Lesson 9
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