Question

what is the relative value of f(x) when the value of x is close to 2 for the function $f(x) = \frac{1}{x - 2}$, whose graph is shown below?
a. a very small number
b. either a very large number or a very small number
c. a very large number
d. zero

Explanation:

Step1: Analyze the function

The function is \( F(x)=\frac{1}{x - 2} \). We need to find the behavior as \( x \) approaches 2.

Step2: Consider left - hand limit (x approaches 2 from the left)

When \( x \) is slightly less than 2 (e.g., \( x=2 - \epsilon \), where \( \epsilon \) is a very small positive number), then \( x - 2=-\epsilon \). So \( F(x)=\frac{1}{-\epsilon}=- \frac{1}{\epsilon} \), which is a very small (negative) number (approaching negative infinity).

Step3: Consider right - hand limit (x approaches 2 from the right)

When \( x \) is slightly greater than 2 (e.g., \( x = 2+\epsilon \), where \( \epsilon \) is a very small positive number), then \( x - 2=\epsilon \). So \( F(x)=\frac{1}{\epsilon} \), which is a very large (positive) number (approaching positive infinity).
So as \( x \) is close to 2, \( F(x) \) can be either a very large number (when approaching from the right) or a very small number (when approaching from the left).

Answer:

B. Either a very large number or a very small number