. . Asymptotes An asymptote for the curve y ( ) is a straight line which is a tangent at ¥ to the curve. In other words the distance between the curve and the straight line tends to zero when the points on the curve approach infinity.
There are three types of asymptotes. They are . Horizontal asymptote , which is parallel to the x -axis. The line y L is said to be a horizontal asymptote for the curve y ( ) if either lim L →+∞ or lim L →−∞ .
Vertical asymptote , which is parallel to the y -axis. The line x is said to be vertical asymptote for the curve y ( ) if lim a f x = ±∞ or lim a f x = ±∞ . . Slant asymptote , A slant (oblique) asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator.
To find the slant asymptote you must divide the numerator by the denominator using either long division or synthetic division. - . - . - .
- - - - - Applications of Differential Calculus Example . Find the asymptotes of the function f x ( ) = . We have, lim → − = −∞ and lim → + = ∞ . Hence, the required vertical asymptote is x = or the y -axis.
As the curve is symmetric with respect to both the axes, y = or the x -axis is also an asymptote. Hence this (rectangular hyperbola) curve has both the vertical and horizontal asymptotes. Example . Find the slant (oblique) asymptote for the function f x ( ) = Since the polynomial in the numerator is a higher degree ( nd ) than the denominator ( st ), we know we have a slant asymptote.
To find it, we must divide the