Question

evaluating a rational function

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$$\begin{array}{|c|c|} \\hline x & f(x) \\\\ \\hline -1 & a \\\\ \\hline -0.1 & b \\\\ \\hline -0.01 & c \\\\ \\hline -0.001 & d \\\\ \\hline \\end{array}$$

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$$\begin{array}{|c|c|} \\hline x & f(x) \\\\ \\hline 0.001 & e \\\\ \\hline 0.01 & f \\\\ \\hline 0.1 & g \\\\ \\hline 1 & h \\\\ \\hline \\end{array}$$

what happens as \\(x\\) approaches zero from the positive direction?
\\(f(x)\\) gets larger

what happens as \\(x\\) approaches zero from the negative direction?

options:

• \\(f(x)\\) goes to zero
• \\(f(x)\\) gets larger
• \\(f(x)\\) gets larger in the negative direction

Explanation:

Analyze the behavior from the positive direction

Using the Numerical Limits and One-Sided Limits knowledge points, we examine the table for positive values of \(x\) as they approach zero:

• For \(x = 1\), \(f(x) = h\)
• For \(x = 0.1\), \(f(x) = g\)
• For \(x = 0.01\), \(f(x) = f\)
• For \(x = 0.001\), \(f(x) = e\)

The first question asks: "What happens as \(x\) approaches zero from the positive direction?"
The selected answer in the image is: "f(x) gets larger". This indicates that as \(x\) decreases towards \(0\) from the right, the values of \(f(x)\) increase significantly (e.g., \(e > f > g > h\)).

Analyze the behavior from the negative direction

Now, we examine the table for negative values of \(x\) as they approach zero:

• For \(x = -1\), \(f(x) = a\)
• For \(x = -0.1\), \(f(x) = b\)
• For \(x = -0.01\), \(f(x) = c\)
• For \(x = -0.001\), \(f(x) = d\)

Since this is a rational function of the form \(f(x) = \frac{1}{x}\) or similar, as \(x\) approaches zero from the negative direction, the values of \(f(x)\) become extremely large in magnitude but negative in sign (e.g., \(-1, -10, -100, -1000\)).

Select the correct option

Looking at the dropdown options for the second question:

• "f(x) goes to zero"
• "f(x) gets larger"
• "f(x) gets larger in the negative direction"

As \(x\) approaches \(0\) from the left (negative direction), the values of \(f(x)\) decrease without bound, which is described as getting larger in the negative direction.

Answer:

• f(x) goes to zero
• f(x) gets larger
• f(x) gets larger in the negative direction (Correct answer)