Asymptotes
An asymptote of a curve y = f (x) that has an infinite branch is called a line such that the distance between the point (x, f (x)) lying on the curve and the line approaches zero as the point moves along the branch to infinity.
Asymptotes can be vertical, oblique (slant) and horizontal. A horizontal asymptote is often considered as a special case of an oblique asymptote.
Vertical Asymptote
The straight line x = a is a vertical asymptote of the graph of the function y = f (x) if at least one of the following conditions is true:
In other words, at least one of the one-sided limits at the point x = a must be equal to infinity.
A vertical asymptote occurs in rational functions at the points when the denominator is zero and the numerator is not equal to zero (i.e. at the points of discontinuity of the second kind). For example, the graph of the function
A function which is continuous on the whole set of real numbers has no vertical asymptotes.
Oblique Asymptote
The straight line
Similarly, we introduce oblique asymptotes as
The oblique asymptotes of the graph of the function
Therefore, when finding oblique (or horizontal) asymptotes, it is a good practice to compute them separately.
The coefficients
A straight line
Proof.
Necessity
Let the straight line
or, equivalently,
Dividing both sides of the equation by
Consequently, in the limit as
Sufficiency
Suppose that there are finite limits
The second limit can be written as
that meets the definition of an oblique asymptote. Thus, the straight line
Note:
Similarly we can prove the theorem for the case of
Horizontal Asymptote
In particular, if
A straight line
The case
Asymptotes of a Parametrically Defined Curve
Let a plane curve be defined by the parametric equations
This curve has a vertical asymptote
Similarly, a parametrically defined curve has a horizontal asymptote
Here
A parametrically defined curve has an oblique (slant) asymptote
and the coefficients
Asymptote of a Polar Curve
Consider a curve given in polar coordinates by the equation
Its asymptote (if it exists) can be described by the two parameters: the distance
These parameters
Notes:
- In the last formula, the limit transition
can be replaced by an equivalent condition - The parameter
can be both positive and negative.
Asymptotes of an Implicit Curve
An implicitly defined algebraic curve is described by the equation
where the left-hand side is a polynomial in the variables
In differential geometry, the following method for finding an oblique asymptote of an algebraic curve is used. Suppose that the asymptote is described by the equation
where the coefficients
To find a vertical asymptote, it is necessary to substitute its equation
A necessary condition for the existence of a vertical asymptote is the absence of the term of the highest degree
The value of the parameter
The above formulas for the asymptotes of an implicit curve are valid if the curve has no singular points at infinity.
Solved Problems
Example 1.
Find the asymptotes of the function
Solution.
When
Hence,
Find the horizontal asymptote. Compute the limit:
Thus, there exists a horizontal asymptote for the curve, and its equation is
The function has no oblique asymptotes. This can be verified by calculating the coefficients
It can be seen that actually we obtained the horizontal asymptote, which has already been defined above.
So, the graph of the function has the vertical asymptote
Example 2.
Find the asymptotes of the function
Solution.
The function is defined for all
Hence,
Find the possible horizontal asymptotes at
We obtained a horizontal asymptote at
Check for oblique (slant) asymptotes by calculating the slope
The same is for
The graph of the function is sketched in Figure
Example 3.
Find the asymptotes of the function
Solution.
We compute the one-sided limits at
The limits are equal to infinity. Therefore,
Next, let us examine the oblique asymptotes:
This shows that instead of an oblique asymptote there is a horizontal asymptote as
A view of the function and its asymptotes is given in Figure
Example 4.
Find the asymptotes of the function
Solution.
The function has a discontinuity at
the straight line
The function does not have a horizontal asymptote because the following limits are infinite:
We write the function as follows:
The term