Question

Find the horizontal asymptote for $\dfrac{{2{x^2}}}{{{x^2} - 4}}$?

Answer 1

Hint: According to given in the question we have to the horizontal asymptote for $\dfrac{{2{x^2}}}{{{x^2} - 4}}$. So, first of all we have to determine if vertical asymptote occurs as the denominator of a rational function tends to 0. To find the equation set the denominator equal to 0.
Now, we have to solve the quadratic expression which is as given in the denominator of the equation $\dfrac{{2{x^2}}}{{{x^2} - 4}}$ and to determine the roots or solve the quadratic expression we have to use the formula which is as mentioned below:

Formula used:
$ \Rightarrow ({a^2} - {b^2}) = (a + b)(a - b)............(A)$
Hence, with the help of the formula (A) we can easily determine the asymptotes which will be vertical asymptotes.
Now, as we all know that horizontal asymptotes can be occurred as,
$ \Rightarrow {\lim _{x \to \infty }},f(x) \to c$ (a constant).
Then divide the given expression with ${x^2}$ in the numerator and denominator of the expression which is $\dfrac{{2{x^2}}}{{{x^2} - 4}}$.
Now, on dividing and subtracting the terms which can be divided and subtracted easily we can determine the horizontal asymptote for the given equation.

Complete step by step solution:
First of all we have to determine if vertical asymptotes occur as the denominator of a rational function tends to 0. To find the equation set the denominator equal to 0.
Now, solve the quadratic expression which is as given in the denominator of the equation $\dfrac{{2{x^2}}}{{{x^2} - 4}}$ and to determine the roots or solve the quadratic expression we have to use the formula (A) which is as mentioned in the solution hint.
$ \Rightarrow {x^2} - 4 = (x + 2)(x - 2)$

As we know that horizontal asymptotes can be occurred as,
$ \Rightarrow {\lim _{x \to \infty }},f(x) \to c$(a constant)
Now, divide the given expression with ${x^2}$ in the numerator and denominator of the expression. Hence,
$ \Rightarrow \dfrac{{\dfrac{{2{x^2}}}{{{x^2}}}}}{{\dfrac{{{x^2}}}{{{x^2}}} - \dfrac{4}{{{x^2}}}}}$
On dividing and subtracting the terms which can be divided and subtracted easily we can determine the horizontal asymptote for the given equation.
$ = \dfrac{2}{{1 - \dfrac{4}{{{x^2}}}}}$
As,
$ \Rightarrow x \to \pm \infty ,f(x) \to \dfrac{2}{{1 - 0}}$
$ \Rightarrow y = 2$ is the asymptote.

Hence, with the help of the formula (A) we have determined the horizontal asymptote for $\dfrac{{2{x^2}}}{{{x^2} - 4}}$ which is $ \Rightarrow y = 2$.

Note:
• To determine the horizontal asymptote it is necessary that we have to determine the roots/zeros of the quadratic expression which is as given in the denominator of the expression $\dfrac{{2{x^2}}}{{{x^2} - 4}}$.
• As we know that horizontal asymptotes can be occurred as, $ \Rightarrow {\lim _{x \to \infty }},f(x) \to c$ hence, we have to substitute the value of $x \to \pm \infty $to determine the required horizontal asymptote.