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A vertical asymptote is a vertical line that the graph of a function approaches but never touches. For the function \( f(x) = \frac{1}{3-x} \), we find the vertical asymptote by setting the denominator equal to zero: \( 3 - x = 0 \). Solving for \( x \), we get \( x = 3 \). This tells us that the function is undefined at \( x = 3 \) and will approach this line as \( x \) gets very close to 3 from either side. You can understand it like this:

• As \( x \to 3^- \), or when approaching 3 from the left, the function \( f(x) \to -\infty \).
• As \( x \to 3^+ \), or when approaching 3 from the right, the function \( f(x) \to +\infty \).

This behavior shows that near \( x = 3 \), the function grows without bound in opposite directions, forming a vertical asymptote.

Horizontal asymptotes show the behavior of a function as it moves towards extremely large positive or negative \( x \) values. For the function \( f(x) = \frac{1}{3-x} \), we need to understand how it behaves when \( x \to \pm \infty \). As \( x \to +\infty \) or \( x \to -\infty \), the expression \( 3-x \) becomes very large and negative, which makes the fraction \( \frac{1}{3-x} \) very small in magnitude. Therefore, we can say:

• \( \lim_{{x \to +\infty}} f(x) = 0 \).
• \( \lim_{{x \to -\infty}} f(x) = 0 \).

This implies that as \( x \) goes far away in either direction, the value of \( f(x) \) approaches zero, creating a horizontal asymptote at \( y = 0 \). Essentially, the graph will get closer and closer to the x-axis as \( x \) increases or decreases without bound.

Understanding limits at infinity helps us describe the end-behavior of functions. It refers to what happens to a function's values as \( x \) becomes very large and moves towards positive or negative infinity. For \( f(x) = \frac{1}{3-x} \), we have:

• \( \lim_{{x \to +\infty}} f(x) = 0 \) since the denominator \( 3-x \) becomes a large negative number.
• \( \lim_{{x \to -\infty}} f(x) = 0 \) as again \( 3-x \) still heads toward a large negative value, making the fraction tiny.

These limits at infinity show that no matter how large or small \( x \) becomes, the value of the function approaches zero. This aligns with the horizontal asymptote of the function being \( y = 0 \). Understanding these limits provides a clearer picture of how functions behave far from the origin, making it easier to grasp their overall behavior without needing to graph them explicitly.