MTH 120 Lab 2: "Closer and Closer"

Consider the function:[br][br][math]\text{f(x) = \frac{x^2-2x-8}{x-4}}[/math]

What is the domain of [math]\text{f(x)}[/math] ?

[math]\text{(-\infty,4)\cup(4,\infty)}[/math] [math]\text{(4,\infty)}[/math] [math]\text{(-\infty,4)}[/math] 4 [math]\text{(-\infty,4)\cup[4,\infty)}[/math] [math]\mathbb{R}[/math] (all real numbers) [math]\text{(-\infty,3)\cup(5,\infty)}[/math]

Numerically/algebraically investigate the behavior of [math]f\left(x\right)[/math] near [math]x=4[/math] by plugging in values close to 4 to the function. You can do this using a calculator or [url=https://www.wolframalpha.com]Wolfram Alpha[/url]. Specific function values are requested in the question below.

List the values of f(3.8), f(3.9), f(3.99), f(4.01), f(4.1), f(4.2).

Based on what you observed in your calculations, what do you think happens on the graph of [math]\text{f(x)}[/math] as [math]\text{x}[/math] gets closer and closer to 4?

Graph the function below by typing it into the left-hand side bar. You can do this simply by typing "y=(x^2-2x-8)/(x-4)" into the small rectangular box and pressing enter.

Graph of f(x)

Does the graph support your conclusions in the previous questions? Explain your reasoning.

Are you surprised by how the graph appears? Try to explain why the graph has the shape it does.

Fill in the blanks: As [math]x[/math] gets closer and closer to 4, [math]f\left(x\right)[/math] gets closer and closer to ___. However, the value of [math]f\left(4\right)[/math] is ___.

nothing, undefined nothing, 6 6, undefined 6,0 6,6 4,4

The Definition of Limit

You have just concluded that the limit of [math]f\left(x\right)[/math] as [math]x[/math] approaches 4 is 6. We write this symbolically as:[br][br][math]\text{\lim_{x\to4} f(x) = 6}[/math]

Function #2

Now let's repeat a similar process for another function -- and this time, we will look not at a specific x value but at what happens as x grows larger and larger.[br][br]Consider the function:[br][br][math]\text{g(x) = \frac{2e^{3x}}{1+e^{3x}}}[/math][br][br]Compute the values of [math]g\left(x\right)[/math] as [math]x[/math] grows large. Try [math]x=[/math] 10, 100, 1000, 10000...

Based on what you observed, what do you think happens to [math]g\left(x\right)[/math] as [math]x[/math] grows larger and larger?

Now graph the function to see if it matches your suspicions. [br][br]Hint: If you type the letter "e" Geogebra understands what you mean. Be careful to enclose the denominator in parentheses so that it doesn't interpret the denominator of the fraction as only the 1.

Graph of g(x).

Does the graph support your conclusions from the previous question? Explain your reasoning.

You should have just concluded that the limit of as approaches infinity is equal to 2. We write this symbolically as:[br][br][math]\text{\lim_{x\to\infty} g(x) = 2}[/math]

Function #3

One more time! Here's another function, this time a rational function. (Rational means polynomial numerator and polynomial denominator. Here they are given in their factored forms).[br][br][math]\text{r(x) = \frac{2x(x-1)(x-7)}{(x-1)(x-3)(x+2)}}[/math]

What are the vertical asymptotes of [math]r\left(x\right)[/math]?

x=7, x=3, x=-2 x=1 x=3, x=-2 x=1, x=3, x=-2 x=0, x=1

What are the horizontal asymptotes of [math]r\left(x\right)[/math]?

none (graph goes to negative infinity) none (graph goes to infinity) x=3 x=2 x=0

Using your knowledge of the asymptotes of this graph, write at least [i]four[/i] limit statements about [math]r\left(x\right)[/math].

Now, graph the function to see if your conclusions are correct.

You have reached the end of this lab activity! Thanks for playing!