To prove that [tex]\(\lim _{x \rightarrow 9} \frac{4}{x} = \frac{4}{9}\)[/tex], we need to evaluate how the function [tex]\(\frac{4}{x}\)[/tex] behaves as [tex]\(x\)[/tex] approaches 9 from both sides (from values slightly less than 9 and values slightly greater than 9).
Let's go through a step-by-step process to understand this limit:
1. Understanding the Function:
- The function we are examining is [tex]\(\frac{4}{x}\)[/tex]. As [tex]\(x\)[/tex] gets closer to 9, we want to see what happens to the value of the function.
2. Substitute Values Close to 9:
- We substitute values of [tex]\(x\)[/tex] that are very close to 9 and observe the output of the function [tex]\(\frac{4}{x}\)[/tex]. We'll look at values slightly less than 9 (e.g., 8.9, 8.99, 8.999, 8.9999) and values slightly greater than 9 (e.g., 9.1, 9.01, 9.001, 9.0001).
3. Evaluate the Function at Values Slightly Less Than 9:
- For [tex]\(x = 8.9\)[/tex], [tex]\(\frac{4}{x} \approx 0.449438202247191\)[/tex]
- For [tex]\(x = 8.99\)[/tex], [tex]\(\frac{4}{x} \approx 0.44493882091212456\)[/tex]
- For [tex]\(x = 8.999\)[/tex], [tex]\(\frac{4}{x} \approx 0.44449383264807196\)[/tex]
- For [tex]\(x = 8.9999\)[/tex], [tex]\(\frac{4}{x} \approx 0.44444938277091967\)[/tex]
As we can see, as [tex]\(x\)[/tex] gets closer to 9 from the left, the value of [tex]\(\frac{4}{x}\)[/tex] gets closer to [tex]\(\frac{4}{9}\)[/tex].
4. Evaluate the Function at Values Slightly Greater Than 9:
- For [tex]\(x = 9.1\)[/tex], [tex]\(\frac{4}{x} \approx 0.43956043956043955\)[/tex]
- For [tex]\(x = 9.01\)[/tex], [tex]\(\frac{4}{x} \approx 0.4439511653718091\)[/tex]
- For [tex]\(x = 9.001\)[/tex], [tex]\(\frac{4}{x} \approx 0.44439506721475397\)[/tex]
- For [tex]\(x = 9.0001\)[/tex], [tex]\(\frac{4}{x} \approx 0.4444395062277086\)[/tex]
Likewise, as [tex]\(x\)[/tex] gets closer to 9 from the right, the value of [tex]\(\frac{4}{x}\)[/tex] approaches [tex]\(\frac{4}{9}\)[/tex].
5. Conclusion:
- By examining the output values from both sides of [tex]\(x = 9\)[/tex], it is evident that as [tex]\(x\)[/tex] approaches 9, the value of the function [tex]\(\frac{4}{x}\)[/tex] approaches [tex]\(\frac{4}{9}\)[/tex].
- Therefore, we can conclude that:
[tex]\[
\lim _{x \rightarrow 9} \frac{4}{x} = \frac{4}{9}
\][/tex]
This completes our proof. The values we computed for different [tex]\(x\)[/tex] values approaching 9 indeed converge to [tex]\(\frac{4}{9}\)[/tex], validating the limit.
