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Rational functions are expressions that consist of fractions with polynomials in both the numerator and the denominator. These are continuous everywhere except where the denominator is zero. When a function is continuous, its limit can be easily found by direct substitution. If plugging in values does not yield a valid number, the function may approach infinity or negative infinity, or it can be undefined at that point. For example, the function \( \frac{x^{3}+4x+8}{2x^{3}-2} \) is a rational function. It becomes undefined if the denominator equals zero at any points, which is why recalculating or simplifying is sometimes necessary when approaching the limit.

An undefined expression occurs when a mathematical operation cannot produce a meaningful result. In the context of limits, undefined expressions generally happen when the denominator of a fraction is zero. When plugging \(x = 1\) into the example function \( \frac{1^{3}+4\times1+8}{2\times1^{3}-2} \), the denominator becomes zero, creating an undefined expression: \( \frac{13}{0} \).
Since division by zero is not possible in mathematical terms, we must check behaviors approaching this point. In limits, we often trace behavior from the positive or negative sides to determine if a limit goes towards infinity, negative infinity, or a particular value. This context guided the solution steps, identifying the function's tendency to infinity as \( x \) approaches 1 from the right.

Algebraic simplification involves rewriting an expression in a simpler form. This process is crucial for understanding limits, especially when the direct substitution leads to undefined forms like \( \frac{0}{0} \) or \( \frac{a}{0} \). Although no further simplification was possible in the provided exercise \( \frac{x^{3}+4x+8}{2x^{3}-2} \), simplifying such expressions is often the next step in problem-solving if possible.
Typically, you might factor polynomials or cancel terms to simplify, but since the function's particulars did not allow these, the solution focused on behavior as \( x \) approached from the right. By checking numerator and denominator signs, the solution deduced that the limit tends towards infinity. Hence, even without simplification, understanding the behavior was key.