47-52 Find the horizontal and vertical asymptotes of each...

47-52 Find the horizontal and vertical asymptotes of each curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the asymptotes.
$$
y=\frac{5+4 x}{x+3}
$$

Key Concepts

Rational Functions

Rational functions are ratios of two polynomial functions. They are important in understanding how the behavior of the numerator and denominator determines properties such as asymptotes and end behavior. Analyzing rational functions involves comparing the degrees of the numerator and denominator and identifying values that make the denominator zero, which are critical for finding asymptotes.

Vertical Asymptote

A vertical asymptote occurs where the function's value increases or decreases without bound as the input approaches a certain value. In the context of rational functions, vertical asymptotes typically arise at points where the denominator is equal to zero (and is not cancelled by a corresponding factor in the numerator), indicating a discontinuity in the function.

Horizontal Asymptote

Horizontal asymptotes describe the behavior of a function as the independent variable approaches positive or negative infinity. For rational functions, the horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients; if the numerator's degree is lower, the asymptote is y = 0; and if higher, the function does not have a horizontal asymptote but may have an oblique asymptote.

End Behavior and Limits

The study of end behavior in rational functions involves taking limits as the independent variable tends toward infinity or negative infinity. These limits help determine horizontal asymptotes by illustrating what value, if any, the function approaches as it moves away from the region near the vertical asymptotes.

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