Find the horizontal, vertical, and slant asymptotes of rational functions.
Rational Function Input
f(x) = (0x² + 1x + 0) / (1x² + 0x + -4)
Analysis Results
Vertical Asymptotesx = 2.00, x = -2.00
Horizontal Asymptotey = 0
Slant (Oblique) AsymptoteNone
Understanding Asymptotes
An asymptote is a line that a graph of a function approaches as the input (x) or the output (y) goes toward infinity. They are critical for understanding the global behavior of functions, especially rational functions with divisions by zero.
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Pro Tips
Vertical asymptotes occur where the denominator is zero (and the numerator is not).
Horizontal asymptotes depend on the degrees of the numerator and denominator.
A slant asymptote occurs if the degree of the numerator is exactly one higher than the denominator.
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Fun Facts
"The term 'asymptote' is derived from the Greek 'asumptotos', which means 'not falling together'."
"Asymptotes are common in physics; for example, terminal velocity is an asymptote for the speed of a falling object."
"The hyperbola y = 1/x is the most famous example of a function with both horizontal and vertical asymptotes."
