Question

How do you find the vertical, horizontal and oblique Asymptote given:
$f(x)=\dfrac{3{{x}^{2}}-2x-1}{x+4}$

Answer 1

In this question we have to find out the vertical, horizontal and oblique Asymptote then firstly find the vertical asymptote by setting the value of denominator equals to zero then find horizontal asymptote by using the fact that when the degree of the numerator is greater than the degree of the denominator and then find out oblique asymptote by dividing the function and you can obtain the given result.

Complete step-by-step solution:
An asymptote is a straight line that approaches a curve indefinitely but does not intersect at any point. In other words, as a curve approaches infinity, it approaches an asymptote line.
Vertical Asymptote:
A vertical asymptote will exist for all rational formulations. Simply explained, when the denominator equals $0$ , a vertical asymptote occurs. In mathematics, an asymptote is simply an undefined point of a function; division by $0$ is also undefined.
Horizontal Asymptote:
In a rational function, there are two conceivable possibilities for a horizontal asymptote. Both are dependent on the numerator and denominator's maximum degrees.
At $y=0$, there will be a horizontal asymptote if the denominator's degree is greater than the numerator and if you do not know then the degree of any function is the highest function belongs to $x$
Oblique Asymptote:
When the degree of the denominator is less than that of the numerator, oblique asymptotes occur. There will be an oblique asymptote at the response to the division of the denominator and the numerator in the rational function $h(x)=\dfrac{a{{x}^{2}}+bx+c}{dx+n}$ To find oblique asymptotes, you'll need to know how to divide polynomials using either long division or synthetic division.
Now according to the question:
We have given $f(x)=\dfrac{3{{x}^{2}}-2x-1}{x+4}$
To find the vertical asymptote we need to set the denominator equals to $0$
$\Rightarrow x+4=0$
$\Rightarrow x=-4$
Hence the vertical asymptote will be $x=-4$
There will be no horizontal asymptote in this function because the denominator is neither equal to nor greater than the numerator in degree.
By synthetic division let:
$\Rightarrow p(x)=3{{x}^{2}}-2x-1$
$\Rightarrow q(x)=x+4$
Now divide the function to get oblique asymptote:
We will get the remainder:
$\Rightarrow 3x-14$
Hence we can conclude that the vertical asymptote is $x=-4$ there is no horizontal asymptote, and oblique asymptote will be $3x-14$.

Note: A curve's asymptote is the line created by the curve's movement and a line moving continuously towards zero. In other words, as a curve approaches infinity, it approaches an asymptote line. Always plot the asymptote graph, remembering to create the axis first, then mark the point and plot the graph.