Overview
A line that a curve approaches is known as asymptote. Any graph (curve) approaches to it but never touches it.
There are total three types of asymptotes-
1. Vertical asymptote
2. Horizontal asymptote
3. Oblique/slant asymptote
Asymptotes
An asymptote of a curve of function y = f(x) is a line which does not intersect on the graph.
Types of asymptotes-
1. Vertical asymptote-
A line x = a which is straight is a vertical asymptote of the graph of the function y = f(x) if atleast one of the following condition does it follow-
2. Horizontal asymptote-
A line y = L is called horizontal asymptote to the curve of the function f(x)-
If f(x)→ L as x→∞ or as x→-∞
Example: Find the horizontal asymptote of the function-
Sol. In order to find the horizontal asymptote-
Hence the horizontal asymptote is y = 2.
3. Oblique asymptote or slant asymptote-
A straight line y = mx + c where m ≠ 0 will be an oblique/slant asymptote to the graph of the function ‘f’ if-
Note- the value of m can be find as-
And the value of c can be find as-
Example: Find the slant asymptote of the function f(x) = x + x .
Sol. Find the value of m-
Hence the y = x + c,
Now find c
Here c must not be infinite.
So we can say that f does not have a slant asymptote at ∞.
Procedure to find the asymptotes parallel to axes of a polynomial function-
Theorem-
Suppose f(x , y) is a polynomial in x and y.
A straight line y = c is an asymptote of a curve f(x , y) = 0 if an only if y – c is a factor of the co-efficient of the highest power of x in f(x , y).
Example: Determine the asymptotes parallel to axes of the curve:
Sol. The given function can be written as-
Asymptote parallel to x-axis-
Equating the coefficient of the highest power of x to zero, we get y² = 0 which means y = 0. This is an asymptote.
Asymptote parallel to y-axis-
Equating the coefficient of the highest power of x to zero, we get x² – a² = 0
Which gives x = ±a , that means x = +a , x = -a
So that these are the asymptotes.
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