Find the vertical asymptotes, if any, of the graph of the rational function. \( f(x)=\frac{x}{x^{2}+4} \)

To determine the vertical asymptotes of the rational function \( f(x) = \frac{x}{x^{2} + 4} \), we need to identify the values of \( x \) that make the denominator equal to zero, provided the numerator is not also zero at those points. 1. **Find when the denominator is zero:** \[ x^{2} + 4 = 0 \] \[ x^{2} = -4 \] \[ x = \pm 2i \] Here, \( i \) is the imaginary unit, representing the square root of -1. 2. **Analyze the solutions:** The solutions \( x = \pm 2i \) are **imaginary numbers**, not real numbers. Vertical asymptotes occur at real values of \( x \) where the function is undefined (i.e., where the denominator is zero but the numerator is not). 3. **Conclusion:** Since there are no real values of \( x \) that make the denominator zero, the function \( f(x) \) does **not** have any vertical asymptotes in the real plane. **Final Answer:** There are no vertical asymptotes for the function \( f(x) = \frac{x}{x^{2} + 4} \).