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Vertical asymptotes occur in a rational function when the denominator equals zero, indicating that the function will approach infinity at that point. In the function \( t(x) = \frac{(x-1)(x-2)}{(x-3)(x-4)} \), vertical asymptotes are found by setting the denominator to zero: \((x-3)(x-4) = 0\).
Solving these equations gives us the potential vertical asymptotes at \(x=3\) and \(x=4\). A critical step is to confirm that these asymptotes are not canceled out by the numerator, meaning they don't have corresponding factors.
In this example, neither \((x-1)\) nor \((x-2)\) in the numerator cancels out the factors \((x-3)\) or \((x-4)\), confirming that there are indeed vertical asymptotes at \(x=3\) and \(x=4\).

• Vertical asymptotes are not influenced by the numerator unless they cancel out.
• It's essential to solve the denominator for potential asymptotes.
• Check for factor cancellation to avoid mistakes.

Horizontal asymptotes provide information about the behavior of a rational function as \(x\) approaches infinity. These asymptotes depend on the degrees of the numerator and denominator polynomials.
For the function \( t(x) = \frac{(x-1)(x-2)}{(x-3)(x-4)} \), both the numerator and the denominator are quadratic polynomials (degree 2).
When the degrees of the numerator and denominator are equal:

• The horizontal asymptote is found at \(y=\frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and denominator, respectively.
• In this rational function, both coefficients are 1, giving a horizontal asymptote at \(y=1\).

If the degree of the numerator were higher:

• No horizontal asymptote would exist.
• The function would likely have an oblique or slant asymptote.

Understanding polynomial degrees is crucial for analyzing rational functions and their asymptotic behavior. The degree of a polynomial is the highest power of the variable in the polynomial expression.
In rational functions:

• The degree of the numerator helps determine both the behavior of the function and any asymptotes it may have.
• The degree of the denominator is key for discovering vertical asymptotes and contributes to the horizontal asymptote analysis.

For \(t(x) = \frac{(x-1)(x-2)}{(x-3)(x-4)}\), both polynomials are quadratic, meaning each has a degree of 2.
Determining the degree is straightforward:

• Expand the expression (if needed) to identify the highest power of \(x\).
• A higher degree in the denominator could indicate more complex vertical asymptotes.
• A higher degree in the numerator can alter the existence or position of a horizontal asymptote.