2.6 HW Problems

Table of Contents:

Finite Limits as $x \rightarrow \pm \infin$

• We use $\infin$ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.
• So for $f(x)=\frac{1}{x}$

$$ \lim_{x \rightarrow \, \infin} \frac{1}{x}=0 $$

$$ \lim_{x \rightarrow \, -\infin} \frac{1}{x}=0 $$

$$ \lim_{x \rightarrow \, 0^+} \frac{1}{x}=\infin $$

$$ \lim_{x \rightarrow \, 0^-} \frac{1}{x}=-\infin $$

3 Cases for Rational Functions $x \rightarrow \pm \infin$

1. Degree of denominator is GREATER than numerator.

   1. answer → ALWAYS 0

   $$ \lim_{x \rightarrow \, \infin} \frac{x^{\color{yellow}2}-x}{x^{\color{green}3}}+1 $$

   $$ \lim_{x \rightarrow \, \infin} \frac{x^2-x}{x^3+1} \rightarrow \frac{\cancel{\frac{x^2}{x^2}}-\frac{x}{x^2}}{\frac{x^3}{x^2}+\frac{1}{x^2}} \rightarrow \frac{1-\xcancel{\frac{1}{x^2}}}{x+\xcancel{\frac{1}{x^2}}} = 0 $$

2. Degree of numerator is GREATER than denominator.

   1. answer → $\infin$ (or $-\infin$ if the x is negative)

      $$ \lim_{x \rightarrow \, \infin} \frac{x^{\color{yellow}3}+x}{x^{\color{green}2}}-x $$

      $$ \lim_{x \rightarrow \, \infin} \frac{x^3+x}{x^2-x} \rightarrow \frac{\frac{x^3}{x^2}+\frac{x}{x^2}}{\cancel{\frac{x^2}{x^2}}-\frac{x}{x^2}} \rightarrow \frac{x+\xcancel{\frac{1}{x}}}{1-\xcancel{\frac{1}{x}}} = \infin $$

3. Degree is the SAME. 1.

    $$
    \\lim_{x \\rightarrow \\, \\infin} \\frac{2x^{\\color{yellow}2}+1}{3x^{\\color{green}2}}-2
    $$
    
    $$
    \\lim_{x \\rightarrow \\, \\infin} \\frac{2x^2+1}{3x^2-2} \\rightarrow \\frac{\\cancel{\\frac{2x^2}{x^2}}+\\frac{1}{x^2}}{\\cancel{\\frac{3x^2}{x^2}}-\\frac{2}{x^2}} \\rightarrow \\frac{2+\\xcancel{\\frac{1}{x^2}}}{3-\\xcancel{\\frac{2}{x^2}}} = \\frac{2}{3}
    $$
   

Interesting Cases…

$$ \lim_{x\rightarrow 0^+} \frac{1}{x^\frac{\color{yellow}{2}}{3}} = \infin $$

$$ \lim_{x\rightarrow 0^-} \frac{1}{x^\frac{\color{yellow}{2}}{3}} = \infin $$

$$ \lim_{x\rightarrow 0} \frac{1}{x^\frac{\color{yellow}{2}}{3}} = \infin $$

$$ \lim_{x \rightarrow 0^+} \frac{1}{x^\frac{\color{yellow}{1}}{3}} = \infin $$

$$ \lim_{x \rightarrow 0^-} \frac{1}{x^\frac{\color{yellow}{1}}{3}} = -\infin $$

$$ \lim_{x \rightarrow 0} \frac{1}{x^\frac{\color{yellow}{1}}{3}} = DNE \, \leftarrow \text{since L and R side isn't the same} $$

• Just focusing on the numerator (1 or 2) using logic, is the numerator is 2 then the answer will always be positive:

$$ (-9)^2=+81 $$

$$ (-1)^2 = +1 $$