Determine all horizontal and vertical asymptotes. For each...

Determine all horizontal and vertical asymptotes. For each vertical asymptote, determine whether $f(x) \rightarrow \infty$ or $f(x) \rightarrow-\infty$ on either side of the asymptote.
$$f(x)=\frac{3 x^{2}+1}{x^{2}-2 x-3}$$

Key Concepts

Rational Functions

Rational functions are functions defined as the ratio of two polynomials. Understanding their properties involves analyzing the degrees of the numerator and denominator, which helps in determining the behavior of the function at extreme values (infinite limits) and near the points where the denominator is zero.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the independent variable approaches positive or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator: if both degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

Vertical Asymptotes

Vertical asymptotes occur at values of the independent variable where the function becomes unbounded, typically where the denominator equals zero (and the zero is not canceled by a corresponding factor in the numerator). In rational functions, identifying these involves solving the equation set by the denominator equal to zero.

One-Sided Limits and Sign Analysis

One-sided limits and sign analysis are used to determine whether a rational function approaches positive or negative infinity near a vertical asymptote. This involves evaluating the sign of the function’s numerator and denominator on either side of the asymptote to understand the directional behavior as the variable approaches the asymptotic value from the left or right.

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