College

We appreciate your visit to Given the function tex f x frac 6x 1 3x 2 tex as the values of tex x tex increase to infinity what happens to. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Given the function [tex]f(x) = \frac{6x - 1}{3x + 2}[/tex], as the values of [tex]x[/tex] increase to infinity, what happens to the values of [tex]f(x)[/tex]?

**Solution:**

Set up a table with increasing values of [tex]x[/tex]. The pattern in the table shows that as [tex]x[/tex] approaches infinity, [tex]f(x)[/tex] approaches 2. It keeps getting closer to 2; it never reaches 2. You can say, "The limit of [tex]f(x)[/tex], as [tex]x[/tex] approaches infinity, is 2," written as [tex]\lim_{x \rightarrow \infty} f(x) = 2[/tex].

[tex]\text{Lim}[/tex] is an abbreviation for limit. The arrow represents "approaching." The symbol for infinity is [tex]\infty[/tex].

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
100 & 1.9 \\
1,000 & 1.9 \\
90,000 & 1.9 \\
900,000 & 1.99 \\
8,000,000 & 1.99 \\
50,000,000 & 1.99 \\
2,000,000,000 & 1.99 \\
\hline
\end{array}
\]

**Check Your Understanding:**

If [tex]f(x) = \frac{1}{x}[/tex], use a table and your calculator to find [tex]\lim_{x \rightarrow \infty} f(x)[/tex].

Answer :

Sure! Let's find the limit of the function [tex]\( f(x) = \frac{1}{x} \)[/tex] as [tex]\( x \)[/tex] approaches infinity. We'll approach this by examining the pattern of the values of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] becomes larger and larger.

### Step-by-Step Solution

1. Choose Large Values for [tex]\( x \)[/tex]:
To understand how [tex]\( f(x) \)[/tex] behaves as [tex]\( x \)[/tex] increases, we select some large numbers for [tex]\( x \)[/tex]. Let's consider:
- [tex]\( x = 100 \)[/tex]
- [tex]\( x = 1000 \)[/tex]
- [tex]\( x = 10,000 \)[/tex]
- [tex]\( x = 100,000 \)[/tex]
- [tex]\( x = 1,000,000 \)[/tex]
- [tex]\( x = 10,000,000 \)[/tex]
- [tex]\( x = 100,000,000 \)[/tex]

2. Calculate [tex]\( f(x) = \frac{1}{x} \)[/tex] for Each [tex]\( x \)[/tex]:
- For [tex]\( x = 100 \)[/tex], [tex]\( f(100) = \frac{1}{100} = 0.01 \)[/tex]
- For [tex]\( x = 1000 \)[/tex], [tex]\( f(1000) = \frac{1}{1000} = 0.001 \)[/tex]
- For [tex]\( x = 10,000 \)[/tex], [tex]\( f(10,000) = \frac{1}{10,000} = 0.0001 \)[/tex]
- For [tex]\( x = 100,000 \)[/tex], [tex]\( f(100,000) = \frac{1}{100,000} = 0.00001 \)[/tex]
- For [tex]\( x = 1,000,000 \)[/tex], [tex]\( f(1,000,000) = \frac{1}{1,000,000} = 0.000001 \)[/tex]
- For [tex]\( x = 10,000,000 \)[/tex], [tex]\( f(10,000,000) = \frac{1}{10,000,000} = 0.0000001 \)[/tex]
- For [tex]\( x = 100,000,000 \)[/tex], [tex]\( f(100,000,000) = \frac{1}{100,000,000} = 0.00000001 \)[/tex]

3. Analyze the Pattern:
As the values of [tex]\( x \)[/tex] increase, the values of [tex]\( f(x) \)[/tex] become smaller and smaller. It becomes apparent that [tex]\( f(x) \)[/tex] is approaching zero.

4. Conclusion:
From this pattern, we conclude that as [tex]\( x \)[/tex] approaches infinity, the function [tex]\( f(x) = \frac{1}{x} \)[/tex] approaches 0. Thus, we can write:
[tex]\[
\lim_{{x \to \infty}} f(x) = 0
\][/tex]

This means that as you choose larger and larger values for [tex]\( x \)[/tex], the function value will get closer and closer to 0.

Thanks for taking the time to read Given the function tex f x frac 6x 1 3x 2 tex as the values of tex x tex increase to infinity what happens to. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada