Limits are fundamental mathematical tools used to determine the behavior of functions as their input approaches certain values or infinity. This powerful concept allows us to identify three types of asymptotes: vertical, horizontal, and oblique asymptotes. Let's explore how limits help us find these important features of functions.
Vertical asymptotes occur when the function approaches positive or negative infinity as x approaches a specific value. To find vertical asymptotes, we look for values where the denominator equals zero but the numerator doesn't. For this function, setting the denominator equal to zero gives us x equals 1 and x equals 3, creating two vertical asymptotes.
Horizontal asymptotes occur when the function approaches a finite value as x approaches positive or negative infinity. For rational functions, we compare the degrees of the numerator and denominator. When both have the same degree, the horizontal asymptote is the ratio of their leading coefficients. Here, both numerator and denominator have degree 2, so the horizontal asymptote is y equals 2.
Oblique or slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find the oblique asymptote, we use polynomial long division or limits. The slope m is found by taking the limit of f(x) over x as x approaches infinity, and the y-intercept b is found by taking the limit of f(x) minus mx. For this function, the oblique asymptote is y equals x plus 1.
In summary, limits are essential tools for finding asymptotes. For vertical asymptotes, we find where the denominator equals zero and check if the limits approach infinity. For horizontal asymptotes, we evaluate limits as x approaches infinity. For oblique asymptotes, we use polynomial division when the numerator's degree exceeds the denominator's by one. Understanding these limit-based techniques allows us to completely analyze function behavior and sketch accurate graphs.