Question

How to find obliquely asymptote of function?​

Answer 1

Answer:

Polynomial Division to Find Oblique Asymptotes

The idea is that when you do polynomial division on a rational function that has one higher degree on top than on the bottom, the result always has the form mx + b + remainder term. Then the oblique asymptote is the linear part, y = mx + b.

Answer 2

Step-by-step explanation:

Question : How to find obliquely asymptote of function?

Ans: There are three types of asymptotes of a function

1. Vertical asymptotes
2. Horizontal asymptotes
3. Oblique asymptotes

Here we are discussed about oblique asymptotes

Definition:

Oblique asymptotes are the non-vertical lines in the form of y=mx+c; where m and c are constant; normally known as slope and y-intercept respectively.

The function reaches either +∞ and -∞ along with these lines.

Points to remember:

1. All the polynomial functions having degree 2 or more have no oblique asymptotes.
2. All the rational function f(x)=p(x)/q(x); q(x)≠0,have no oblique asymptotes if p(x) and q(x) have same degree.
3. Hyperbolas also have two oblique asymptotes.

How to find oblique asymptotes:

1) If function is in the form of rational function,having degree of numerator one more than degree of denominator,then apply division.

The quotient of division is oblique asymptote of the function.

For eg:

$f(x) = \frac{ {x}^{3} + 3 {x}^{2} + 3x + 5 }{ {x}^{2} + 2x + 1 } \\ \\{x}^{2} + 2x + 1) \: {x}^{3} + 3 {x}^{2} + 3x + 5 \: (x + 1\\ \: \: \: \: \: \: \: \: \: \: {x}^{3} + 2 {x}^{2} + x \\ \: \: \: \: \: \: \: \: \: ( - ) \: \: ( - ) \: \: ( - ) \: \\ \: \: \: \: \: \: \: \: \: - - - - - - - - \\ {x}^{2} + 2x + 5 \\ {x}^{2} + 2x + 1 \\ ( - ) \: ( - ) \: ( - ) \\ - - - - - - - - \\ 4 \\ - - - - - - - - -$

Thus,

$\bold{y = x + 1}$

is the oblique asymptote of f(x).

2) If hyperbola is represented by the equation

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

then its oblique asymptotes are given by

$\bold{y=\left(\frac{b}{a}\right)x}\:\:and\:\:\bold{y=\left(\frac{-b}{a}\right)x}$

Hope it helps you.