Identify holes and asymptotes of rational functions

Factoring a rational function reveals that a common factor in the numerator and denominator produces a removable discontinuity, or hole, at the corresponding excluded input; the hole’s coordinates are found by evaluating the reduced function at that input. Zeros of the remaining denominator factors identify vertical asymptotes, while comparing numerator and denominator degrees determines horizontal or, when the numerator is exactly one degree higher, slant asymptotes and describes end behavior; more advanced asymptotic analysis is outside this scope.

Detailed Explanation: Identify holes and asymptotes of rational functions

To identify holes and asymptotes, first factor the numerator and denominator. Then look for common factors.

Consider

f(x)=\frac{x^2-1}{x^2-3x+2}.

1. Factor completely

Factor the numerator and denominator:

x^2-1=(x-1)(x+1)

and

x^2-3x+2=(x-1)(x-2).

So,

f(x)=\frac{(x-1)(x+1)}{(x-1)(x-2)}.

The original denominator is zero when $x=1$ or $x=2$ , so both values are excluded from the domain.

2. Find the hole

The common factor $(x-1)$ cancels:

f(x)=\frac{x+1}{x-2}, \qquad x\ne 1,2.

Because $(x-1)$ was canceled, $x=1$ produces a hole.

To find the hole’s $y$ -coordinate, substitute $x=1$ into the reduced function:

y=\frac{1+1}{1-2}=\frac{2}{-1}=-2.

Therefore, the hole is

\boxed{(1,-2)}.

3. Find the vertical asymptote

The denominator that remains after cancellation is $x-2$ . Set it equal to zero:

x-2=0

so the vertical asymptote is

\boxed{x=2}.

The canceled factor $(x-1)$ does not give a vertical asymptote; it gives the hole.

4. Find the horizontal asymptote

In the original function, the numerator and denominator both have degree $2$ . When the degrees are equal, divide the leading coefficients:

\frac{1}{1}=1.

Thus, the horizontal asymptote is

\boxed{y=1}.

This also describes the end behavior: as $x$ becomes very large positive or very large negative, the graph approaches $y=1$ .

For this function:

Hole: $\boxed{(1,-2)}$
Vertical asymptote: $\boxed{x=2}$
Horizontal asymptote: $\boxed{y=1}$

Learn by doing: Identify holes and asymptotes of rational functions

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Rational Functions and Asymptotes - Calculate Horizontal Asymptote (Factored)

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