Loading [MathJax]/jax/element/mml/optable/BasicLatin.js

Indeterminate Limits—The Factorable 00 Form

Download this web page as a pdf with answer key

If the limit of a rational function produces a 00 form...

  1. factor the numerator and denominator,
  2. divide out the common factor(s),
  3. then re-evaluate the limit .

Examples

The Limit Exists

Example 1

Evaluate lim

Step 1

Confirm that the limit has an indeterminate form.

\displaystyle \lim_{x\to -3}\frac{x^2+x-6}{x^2+8x+15} = \frac{(-3)^2+(-3) - 6}{(-3)^2+8(-3)+15} = \color{red}{ \frac 0 0}

Since \frac{0}{0} is an indeterminate form, the limit may (or may not) exist. We have more work to do.

Step 2

Since the function is rational, we can try factoring both the numerator and denominator to identify common factors.

\begin{align*}% \lim_{x\to-3}\frac{x^2+x-6}{x^2+8x+15} % & = \lim_{x\to-3}\frac{{\color{blue}(x+3)}(x-2)}{{\color{blue}(x+3)}(x+5)} \\ % & = \lim_{x\to-3}\frac{x-2}{x+5} \end{align*}

Step 3

Evaluate the simpler limit .

\displaystyle \lim_{x\to-3}\frac{x-2}{x+5} = \frac{-3-2}{-3+5} = -\frac 5 2

Anwser

\displaystyle \lim_{x\to-3}\frac{x^2+x-6}{x^2+8x+15} = -\frac 5 2

The Limit Doesn’t Exist

Example 2

Evaluate: \displaystyle \lim_{x\to4}\frac{2x^2-7x-4}{x^3-8x^2+16x}

Step 1

Confirm the limit has an indeterminate form.

\lim_{x\to4}\frac{2x^2-7x-4}{x^3-8x^2+16x} = \frac{2(4)^2-7(4)-4}{(4)^3-8(4)^2+16(4)} = \color{red}{ \frac 0 0}

Since \frac{0}{0} is an indeterminate form, the limit may (or may not) exist. We have more work to do.

Step 2

Since the function is rational, try factoring to find any common factors.

\begin{align*} \lim_{x\to4}\frac{2x^2-7x-4}{x^3-8x^2+16x} & = \lim_{x\to4}\frac{{\color{blue}(x-4)}(2x+1)}{x{\color{blue}(x-4)}(x-4)} \\ % & = \lim_{x\to4}\frac{2x+1}{x(x-4)} \end{align*}

Step 3

Evaluate the simpler limit .

\lim_{x\to4}\frac{2x+1}{x(x-4)} = \frac{2(4) + 1}{4(4-4)} = \frac 9 0

Anwser

\displaystyle \lim_{x\to4}\,\frac{2x^2-7x-4}{x^3-8x^2+16x} does not exist.

Practice Problems

Problem 1

Evaluate \displaystyle \lim_{x\to 2} \frac{x^2+5x-14}{x^2-4}

Problem 2

Evaluate \displaystyle \lim_{x\to -1} \frac{x^2-4x-5}{x^2+10x+9}

Problem 3

Evaluate \displaystyle \lim_{x\to\frac 1 3} \frac{3x^2-7x+2}{3x^2+5x-2}

Problem 4

Evaluate \displaystyle \lim_{x\to-8} \frac{2x^2+19x+24}{2x^2+15x-8}

Problem 5

Evaluate \displaystyle \lim_{x\to6} \frac{x^2-3x-18}{x^2-12x+36}

Problem 6

Evaluate \displaystyle \lim_{x\to-5}\,% \frac{2x^2+13x+15}{x^3+10x^2+25x}%

Problem 7

Evaluate \displaystyle \lim_{x\to -\frac 2 5}\,% \frac{15x^3+x^2-2x} {25x^2 + 20x + 4}

Download this web page as a pdf with answer key

back toLimits of Piecewise-Defined Functions next to Indeterminate Limits-Rationalizing

Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!