Limits and Continuity

Lesson 8

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?

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

Target Task

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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?