To answer this question, we need to analyze the behavior of the function [tex]\( f(x) = \frac{2}{x-2} \)[/tex] when [tex]\( x \)[/tex] is close to 2. Specifically, what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches the value of 2.
The function [tex]\( f(x) = \frac{2}{x-2} \)[/tex] contains a denominator that gets very small as [tex]\( x \)[/tex] gets close to 2. This small denominator will significantly affect the value of the overall function. Let's consider each option to see if it makes sense given this behavior.
1. Option A: 2
- For [tex]\( f(x) \)[/tex] to be 2, we must solve the equation:
[tex]\[ \frac{2}{x-2} = 2 \][/tex]
Simplifying this gives:
[tex]\[ x - 2 = 1 \][/tex]
[tex]\[ x = 3 \][/tex]
However, 3 is not close to 2. Thus, this option could not be a value of [tex]\( f(x) \)[/tex] when [tex]\( x \)[/tex] is close to 2.
2. Option B: -0.01
- For [tex]\( f(x) \)[/tex] to be -0.01, we must solve the equation:
[tex]\[ \frac{2}{x-2} = -0.01 \][/tex]
Simplifying this gives:
[tex]\[ x - 2 = -200 \][/tex]
[tex]\[ x = -198 \][/tex]
Again, -198 is not close to 2. Therefore, this option is also not feasible.
3. Option C: 10,000
- For [tex]\( f(x) \)[/tex] to be 10,000, we must solve the equation:
[tex]\[ \frac{2}{x-2} = 10,000 \][/tex]
Simplifying this gives:
[tex]\[ x - 2 = \frac{2}{10,000} \][/tex]
[tex]\[ x - 2 = 0.0002 \][/tex]
[tex]\[ x = 2.0002 \][/tex]
This value, 2.0002, is very close to 2. Therefore, it is plausible that [tex]\( f(x) \)[/tex] could be 10,000 when [tex]\( x \)[/tex] is close to 2.
4. Option D: 0.01
- For [tex]\( f(x) \)[/tex] to be 0.01, we must solve the equation:
[tex]\[ \frac{2}{x-2} = 0.01 \][/tex]
Simplifying this gives:
[tex]\[ x - 2 = 200 \][/tex]
[tex]\[ x = 202 \][/tex]
This value, 202, is not close to 2. Hence, this option is not reasonable.
Given this analysis, the most plausible value of [tex]\( f(x) \)[/tex] when [tex]\( x \)[/tex] is close to 2 is given by Option C: 10,000.
