How to Find Vertical Asymptotes | Vertical, Horizontal Asymptote

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How to find vertical Asymptotes. So Asymptote – is the word that most involve around this, find this type of problem is impossible to crack through.

Let’s say Nothing is Impossible…. with a thorough knowledge of the concept and well-learned techniques up in your sleeves (no, you are not a magician, but a thinker), it is possible to find the asymptotes of a function.

An asymptote of a curve is a line that defines the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. 

The word asymptote is derived from the Greek (asymptotes), that means “not falling together”.  The term was introduced by Apollonius of Perga in his work and thesis on conic sections, he used it to mean any line that does not intersect the given curve.

Here we are dealing with 3 types of Asymptotes…. horizontal, vertical and oblique.   For curves given by the graph of a function y = f(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to + infinity or – infinity.  They are the vertical lines near which the function grows without bound.  An oblique asymptote has a slope that is non – zero but finite.

Image of a vertical asymptote:

HOW TO FIND VERTICAL ASYMPTOTES USING VERTICAL METHOD

Image of a horizontal asymptote:

HOW TO FIND VERTICAL ASYMPTOTES USING HORIZONTAL METHOD

Image of an oblique asymptote:

HOW TO FIND VERTICAL ASYMPTOTES USING oblique METHOD

The curve can approach from any side (such as from above or below for a horizontal asymptote), or may actually cross over (possibly many times) and even more away and back again.

HOW TO FIND VERTICAL ASYMPTOTES USING CURve

The important point is that – the distance between the curve and the asymptote tends to zero as they head to infinity (or – infinity).

Asymptotes convey information about the behaviour of curves in the large.  The study of asymptotes of functions forms a part of the subject of asymptotic analysis.

 So, now the question is how to identify the Vertical asymptote of a function.

And the most imperative quest is why Vertical asymptote is relatively important than other 2 types of Asymptotes…. isn’t it inspiring?

The fact is that a graph can have an infinite number of Vertical Asymptotes, but it can only have at most two horizontal asymptotes……so why this response is important to all the seekers?  Let’s explore this further….

Vertical asymptotes are also called the vertical lines that correspond to the zeroes of the denominator of a rational function. 

So basically Vertical asymptotes are straight lines of the equation, towards which a function f(x) approaches infinitesimally closely, but never reaches the line, as f(x) increases without binding.  For these values of x, the function is either unbounded or is undefined.

So, to find Vertical asymptotes, solve the equation n(x) = 0, where n(x) is the denominator of the function.  Remember, in this equation numerator t(x) is not zero for the same x value.  The graph has a vertical asymptote with the equation of x = 1.

The Vertical asymptotes occur at singularities or points at which the rational function is not defined.

How to find Vertical Asymptotes numerically

Find the vertical asymptote:

f(x) = 3x – 1 / x + 5

To solve it, set the denominator to zero.

x + 5 = 0

x = -5

So the vertical asymptote at x is -5.

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