Understanding Oblique Asymptotes | Exploring Slanted Lines in Mathematical Functions

Oblique asymptote

An oblique asymptote, also known as a slant asymptote, is a type of asymptote that occurs when a function approaches a line, either increasing or decreasing, as the x-values tend towards positive or negative infinity

An oblique asymptote, also known as a slant asymptote, is a type of asymptote that occurs when a function approaches a line, either increasing or decreasing, as the x-values tend towards positive or negative infinity. Unlike vertical or horizontal asymptotes, which are typically straight lines, oblique asymptotes have a slanted or tilted appearance.

To determine whether a function has an oblique asymptote, you need to examine the behavior of the function as x approaches infinity or negative infinity. First, divide the function by the leading term in the highest power of x. If the result is a non-zero constant, then this constant will represent the slope of the oblique asymptote. Next, perform long division to determine the quotient of the division.

For example, let’s consider the function f(x) = (3x^2 + 4x + 2) / (x – 1).

Dividing the numerator (3x^2 + 4x + 2) by the denominator (x – 1), we get:

3x + 7
+ ______
x – 1 | 3x^2 + 4x + 2

The quotient is 3x + 7. Since the quotient is not a constant, we can conclude that the function has an oblique asymptote. The slope of the oblique asymptote is the coefficient of the leading term in the quotient, which is 3 in this case.

To find the equation of the oblique asymptote, we can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, the equation of the oblique asymptote would be y = 3x + b. To find the value of b, we need to consider the remainder of the long division. Since the remainder is 7, the y-intercept of the oblique asymptote is b = 7. Thus, the equation of the oblique asymptote is y = 3x + 7.

It’s important to note that an oblique asymptote only exists if the degree of the numerator is exactly one more than the degree of the denominator. If the degrees are not equal, the function may have a vertical asymptote instead.

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