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In calculus, a vertical asymptote occurs in a graph of a function when there is a point where the function grows indefinitely. In simpler words, the function has no finite value at this point, usually because it results in division by zero. For the function \(f(x)=\frac{x^{2}+1}{x+1}\), the denominator becomes zero at \(x=-1\). This means the function might have a vertical asymptote at this point.
However, not all points where the denominator is zero result in a vertical asymptote. To determine whether \(x=-1\) is indeed a vertical asymptote, you must simplify the function to check if a straightforward form persists. In this case, further examination showed that instead of a vertical asymptote, there is a removable discontinuity. This is a clue that the function may just need a bit of algebraic simplification.

A removable discontinuity occurs when a function is undefined at a particular point, but this undefined point can be resolved by simplifying the function. It is like a "hole" in the graph that can be "filled" by redefining the function at that point. For the function \(f(x)=\frac{x^{2}+1}{x+1}\), analyzing the point \(x=-1\) gives you the indeterminate form \(\frac{0}{0}\). This often signals a removable discontinuity.
By rewriting and simplifying the function, we found that we could cancel the common factor \((x+1)\) in both the numerator and the denominator. The revised function \(f(x) = (x-1) + \frac{2}{x+1}\) is continuous, except at \(x=-1\). Here, it becomes apparent that instead of a vertical asymptote, the function merely has a removable discontinuity, which can be handled easily.

Function simplification is a key process in calculus that involves rewriting a complex function into a simpler form. This makes it easier to evaluate, analyze and understand the behavior of the function. For \(f(x)=\frac{x^{2}+1}{x+1}\), simplification involves factoring the numerator, allowing for cancellation with the common factors in the denominator.
The original expression \(x^2 + 1\) in the numerator can be rewritten using basic algebraic techniques. It simplifies to \((x-1)(x+1) + 2\). This simplification helps you identify that \(x+1\) is a common factor, which when removed leads you to the simplified form \(f(x) = (x-1) + \frac{2}{x+1}\). Simplification exposes the "true" behavior of the function, making features like removable discontinuities and finite limits more discernible.

A graphing utility is a tool or software that helps visualize mathematical functions. It plots points on a graph to provide a visual representation of the function's behavior. Using a graphing utility to plot \(f(x)=\frac{x^{2}+1}{x+1}\), especially around critical points like \(x=-1\), can confirm your algebraic conclusions.
Graphing utilities provide various helpful features:

• Visual representation of functions
• Identifying critical points and discontinuities
• Testing conclusions from algebraic simplifications

When you graph the function \(f(x)=\frac{x^{2}+1}{x+1}\), you will observe that the graph is smooth at \(x=-1\), indicating a removable discontinuity. It verifies that there is no abrupt break or infinite value at this point, as would happen with a vertical asymptote. Graphing utilities reinforce the simplicity and reliability of algebraic methods in understanding function behavior.