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

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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 $$ f(x) = \frac{x}{x^{2} + 4} $$.
There are no vertical asymptotes for the function $$ f(x) = \frac{x}{x^{2} + 4} $$.

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

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