Define Vertical and Horizontal Asymptotes in Calculus

What is an Asymptote in Calculus

In mathematical terms, an asymptote is a curve that approaches infinity, asymptotes are lines that appear parallel to it.

What is an Asymptote in Calculus
Asymptote in Math

Types of Asymptotes in Calculus

There are three types of Asymptote in math.

1. Horizontal Asymptote
2. Vertical Asymptote
3. Oblique Asymptote

Difference between Horizontal Vertical and Oblique Asymptotes

Difference between Horizontal Vertical and Oblique Asymptotes

How to find Horizontal Asymptote of Exponential Function?

How to find Horizontal Asymptote of Exponential Function?
It is a Horizontal Asymptote

Horizontal Asymptote: A horizontal asymptote occurs when A curve approaches some constant value b as x approaches infinity (or *infinity)..

How to find Vertical Asymptote of Exponential Function

How to find Vertical Asymptote of Exponential Function
It is a Vertical Asymptote

Vertical Asymptote: A horizontal asymptote occurs when a function of X approaches some constant, c (from the left or right) then the curve continues to infinity (or −infinity).

What is Oblique Asymptote?

What is Oblique Asymptote?
Oblique Asymptote

Oblique Asymptote: A Oblique Asymptote occur when, as x goes to infinity (or −infinity) the curve then becomes a line y=mx+b

Asymptote for a Curve Definition in Math

Definition: A straight line l is called an asymptote for a curve C if the distance between l and C approaches zero as the distance moved along l (from some fixed point on l ) tends to infinity.

The curve C can approach asymptote l as one moves along l in one direction, or in the opposite direction, or in both directions.

How to Find Equation of Horizontal Asymptote?

Suppose the equation y\;=\;f(x) of C is such that y is real and y\rightarrow a as X\rightarrow\infty or X\rightarrow-\infty then y\;=\;a is a Horizontal asymptote. For, the distance between the curve and the straight line y\;=\;a is y\;-\;a and this approaches zero as X\rightarrow\infty or X\rightarrow-\infty.

How to Find Equation of Vertical Asymptote?

If the equation of C is such that y is real and Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a from one side then the straight line x = a is a vertical asymptote.

To see this, observe that (1) x – a is the distance between the curve and the straight line and that this distance is supposed to approach zero (2) Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a , so that

\lim_{y\rightarrow\pm\infty}\;(x-a)\;=\;\lim_{x\rightarrow a}\;(x-a)\;=\;0

Thus, to locate vertical asymptotes we have to find a number a such that \lim_{x\rightarrow a}\;y\;=\;\infty\;or\;-\infty

Similarly, if y\rightarrow mx\;+\;c as x\rightarrow\infty or x\rightarrow-\infty then y\;=\;mx\;+\;c is an asymptotes ( which is neither vertical nor horizontal).

Thus, we inquire for \lim_{x\rightarrow\pm\infty}\;y is studying asymptotes.

Find Horizontal and Vertical asymptote for Algebraic equations

For Algebraic equations, we can find horizontal and vertical asymptotes as follows:

For horizontal asymptotes we write the given equation in the form x\;=\;\frac{\psi(y)}{\theta(y)} and consider those values of y for which \theta(y)\;=\;0.

Similarly, to find a verticle asyptote, we write the given equation in the form y\;=\frac{f(x)}{g(x)} and consider those value of x for which g(x) = 0.

Working Rule for Asymptotes Parallel to the Axes

In an equation of a curve, the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x-axis (respectively y-axis)

Leave a Comment