Questions: R(x) = x^2 / (x^2 + 9) Find Vertical Asymptote

Transcript text: $R(x)=\frac{x^{2}}{x^{2}+9}$ Find Vertical Asymptote

Solution

Solution Steps

To find the vertical asymptote of the rational function $ R(x) = \frac{x^2}{x^2 + 9} $, we need to determine where the denominator is equal to zero, as these are the points where the function is undefined and may have vertical asymptotes.

Solution Approach

Set the denominator equal to zero and solve for $ x $.
Check if the numerator is non-zero at these points to confirm the presence of a vertical asymptote.

Step 1: Identify the Denominator

The function given is

\[ R(x) = \frac{x^2}{x^2 + 9} \]

To find the vertical asymptotes, we need to analyze the denominator, which is

\[ x^2 + 9. \]

Step 2: Set the Denominator to Zero

We set the denominator equal to zero to find the points where the function is undefined:

\[ x^2 + 9 = 0. \]

Step 3: Solve for $ x $

Solving the equation $ x^2 + 9 = 0 $ gives:

\[ x^2 = -9. \]

Taking the square root of both sides results in:

\[ x = \pm 3i. \]

Step 4: Conclusion on Vertical Asymptotes

Since the solutions $ x = 3i $ and $ x = -3i $ are complex numbers, there are no vertical asymptotes in the real number system for the function $ R(x) $.

Final Answer

There are no vertical asymptotes in the real number system. Thus, the answer is

$\boxed{\text{No vertical asymptotes}}$.

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