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Asymptotes are crucial to understanding the behavior of a graph as it approaches specific points or infinity. In the example function \(f(x)=\frac{x}{1+x^{2}}\), we note that a vertical asymptote would occur if the denominator equaled zero. However, given that \(x^{2}\) is never negative, the denominator \(1+x^{2}\) will never be zero, and therefore a vertical asymptote does not exist for this function.

On the other hand, horizontal asymptotes are observed by considering the limits of the function as \(x\) approaches positive or negative infinity. Since both limits lead to the value zero, we identify the \(x\)-axis, or \(y=0\), as the horizontal asymptote. This horizontal asymptote signifies that as \(x\) grows larger in either direction, the value of \(f(x)\) approaches zero.

Critical points of a function are the values where its derivative is zero or undefined. These points indicate where a function switches from increasing to decreasing, or vice versa, and are crucial in sketching the overall shape of the graph. For our function, the first derivative is determined to be \(f'(x)=\frac{1-x^{2}}{(1+x^{2})^{2}}\).

Setting the derivative equal to zero, \(\frac{1-x^{2}}{(1+x^{2})^{2}}=0\), we find the critical points at \(x=-1\) and \(x=1\). Analyzing around these points, we see the function's behavior changes - it decreases up to \(x=-1\), increases between \(x=-1\) and \(x=1\), and decreases again after \(x=1\).

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input variable. Derivatives are fundamental to calculus and provide vital information about the function, including its increasing and decreasing behavior as well as the presence of any maxima or minima.

In the context of graphing the function \(f(x)=\frac{x}{1+x^{2}}\), the derivative \(f'(x)\) reveals where the function's slope is flat (critical points), indicating potential points of inflection or local extrema. The calculation of the derivative and its analysis help us refine our graph and predict the curvature at different regions.

Limits at infinity provide insight into the behavior of functions as the input variable grows without bound in the positive or negative direction. They highlight the long-term trends of the function and are especially crucial in identifying horizontal asymptotes.

For \(f(x)=\frac{x}{1+x^{2}}\), the limits as \(x\) approaches \(\infty\) and \(-\infty\) are both zero, confirming that the function's values get closer and closer to the \(x\)-axis but never actually reach it. This understanding helps in visualizing the end behavior of the graph, ensuring that as we move towards infinity in either direction on the \(x\)-axis, the function approaches zero without touching or crossing the horizontal asymptote.