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In graphing a rational function like \( y = \frac{1}{x-1} \), identifying vertical asymptotes is crucial. A vertical asymptote appears when the denominator of the rational expression equals zero. For our function, the denominator is \( x-1 \). To find where the vertical asymptote occurs, set the denominator to zero and solve for \( x \). This gives \( x - 1 = 0 \), so \( x = 1 \).
The vertical asymptote at \( x = 1 \) means that as \( x \) nears 1 from the left (\( x \to 1^- \)), the value of \( y \) will tend toward negative infinity. Conversely, as \( x \) approaches 1 from the right (\( x \to 1^+ \)), \( y \) will head toward positive infinity.

• Vertical asymptotes are represented as vertical dashed lines on graphs.
• They indicate values \( x \) cannot actually reach due to division by zero.

Understanding vertical asymptotes helps predict the graph's behavior as it approaches these undefined points.

While dealing with rational functions, horizontal asymptotes tell us how the graph behaves as \( x \) approaches very large or very small values. In our rational function \( y = \frac{1}{x-1} \), the horizontal asymptote can be misunderstood. Initially, it looks like as \( x \to \pm\infty \), \( y \) heads to zero, suggesting a horizontal asymptote at \( y = 0 \).
However, in simpler rational functions, this line often acts more as a conceptual boundary rather than a true horizontal asymptote.

• Horizontal asymptotes occur when a function approaches a line as \( x \) becomes very large or small, but never actually reaches it.
• If the degrees of the numerator and denominator are equal, the ratio of their leading coefficients gives the horizontal asymptote.
• If the denominator's degree exceeds the numerator's, the horizontal asymptote is \( y = 0 \).

This particular function, however, reveals more of an oblique asymptote behavior at \( y = x \). Learning to differentiate these subtleties is key to accurately sketching rational functions.

In any rational function like \( y = \frac{1}{x-1} \), the dominant term is what largely dictates the function's end behavior. By identifying this dominant term, which refers to the most influential components when \( x \) is extremely large or small, we understand the overall graph behavior.
For this function, \( \frac{1}{x-1} \) is itself the dominant term as \( x \to \pm\infty \). This means its impact overshadows minor elements, directing the general flow of the curve.

• The dominant term is the part of the function that has the most influence for large values of \( x \).
• It helps understand trends in the graph when \( x \) approaches extreme values.
• Graphically, dominance is reflected in how the curve approaches but never reaches specific lines.

Focusing on the dominant term allows for correct graph plotting, ensuring function trends are accurately captured, even at infinity.