4. vertical and horizontal asymptotes\nfind the vertical asymptotes for the following functions and also…

4. vertical and horizontal asymptotes\nfind the vertical asymptotes for the following functions and also determine what value they approach as (x\rightarrowpminfty).\n1. (f(x)=\frac{1}{x^{2}+4x - 21})\n2. (f(x)=\frac{1}{x^{2}-1})\n3. (f(x)=\frac{x}{3x^{2}+13x - 10})\n4. (f(x)=\frac{x^{2}}{12x^{2}+5x - 2})\nalso determine if the function approaches (pminfty) as the vertical asymptote is approached from the right and from the left.

4. vertical and horizontal asymptotes\nfind the vertical asymptotes for the following functions and also determine what value they approach as (x\rightarrowpminfty).\n1. (f(x)=\frac{1}{x^{2}+4x - 21})\n2. (f(x)=\frac{1}{x^{2}-1})\n3. (f(x)=\frac{x}{3x^{2}+13x - 10})\n4. (f(x)=\frac{x^{2}}{12x^{2}+5x - 2})\nalso determine if the function approaches (pminfty) as the vertical asymptote is approached from the right and from the left.

Answer

Explanation:

Step1: Recall vertical - asymptote condition

Vertical asymptotes occur where the denominator of a rational function is zero. For a rational function $y = \frac{f(x)}{g(x)}$, we set $g(x)=0$.

Step2: Recall horizontal - asymptote rules

  1. If the degree of the numerator $n$ and the degree of the denominator $m$ of a rational function $y=\frac{a_nx^n+\cdots}{b_mx^m+\cdots}$:
    • If $n < m$, the horizontal asymptote is $y = 0$.
    • If $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$.
    • If $n>m$, there is no horizontal asymptote.

Step3: Solve for vertical asymptotes of $f(x)=\frac{1}{x^{2}+4x - 21}$

Set the denominator equal to zero: $x^{2}+4x - 21=(x + 7)(x - 3)=0$. So $x=-7$ and $x = 3$ are the vertical asymptotes. Since the degree of the numerator is $0$ and the degree of the denominator is $2$ ($n = 0,m = 2$), the horizontal asymptote is $y = 0$.

Step4: Solve for vertical asymptotes of $f(x)=\frac{1}{x^{2}-1}$

Set the denominator equal to zero: $x^{2}-1=(x + 1)(x - 1)=0$. So $x=-1$ and $x = 1$ are the vertical asymptotes. Since the degree of the numerator is $0$ and the degree of the denominator is $2$ ($n = 0,m = 2$), the horizontal asymptote is $y = 0$.

Step5: Solve for vertical asymptotes of $f(x)=\frac{x}{3x^{2}+13x - 10}$

Set the denominator equal to zero: $3x^{2}+13x - 10=(3x - 2)(x+5)=0$. So $x=\frac{2}{3}$ and $x=-5$ are the vertical asymptotes. Since the degree of the numerator is $1$ and the degree of the denominator is $2$ ($n = 1,m = 2$), the horizontal asymptote is $y = 0$.

Step6: Solve for vertical asymptotes of $f(x)=\frac{x^{2}}{12x^{2}+5x - 2}$

Set the denominator equal to zero: $12x^{2}+5x - 2=(4x - 1)(3x+2)=0$. So $x=\frac{1}{4}$ and $x=-\frac{2}{3}$ are the vertical asymptotes. Since the degree of the numerator is $2$ and the degree of the denominator is $2$ ($n = 2,m = 2$), and $a_n = 1,b_m=12$, the horizontal asymptote is $y=\frac{1}{12}$.

  1. For $f(x)=\frac{1}{x^{2}+4x - 21}$:
    • Vertical asymptotes: $x=-7,x = 3$
    • Horizontal asymptote: $y = 0$
    • As $x\rightarrow - 7^{-},f(x)\rightarrow-\infty$; as $x\rightarrow - 7^{+},f(x)\rightarrow+\infty$; as $x\rightarrow3^{-},f(x)\rightarrow-\infty$; as $x\rightarrow3^{+},f(x)\rightarrow+\infty$
  2. For $f(x)=\frac{1}{x^{2}-1}$:
    • Vertical asymptotes: $x=-1,x = 1$
    • Horizontal asymptote: $y = 0$
    • As $x\rightarrow - 1^{-},f(x)\rightarrow-\infty$; as $x\rightarrow - 1^{+},f(x)\rightarrow+\infty$; as $x\rightarrow1^{-},f(x)\rightarrow-\infty$; as $x\rightarrow1^{+},f(x)\rightarrow+\infty$
  3. For $f(x)=\frac{x}{3x^{2}+13x - 10}$:
    • Vertical asymptotes: $x=\frac{2}{3},x=-5$
    • Horizontal asymptote: $y = 0$
    • As $x\rightarrow\frac{2}{3}^{-},f(x)\rightarrow-\infty$; as $x\rightarrow\frac{2}{3}^{+},f(x)\rightarrow+\infty$; as $x\rightarrow - 5^{-},f(x)\rightarrow+\infty$; as $x\rightarrow - 5^{+},f(x)\rightarrow-\infty$
  4. For $f(x)=\frac{x^{2}}{12x^{2}+5x - 2}$:
    • Vertical asymptotes: $x=\frac{1}{4},x=-\frac{2}{3}$
    • Horizontal asymptote: $y=\frac{1}{12}$
    • As $x\rightarrow\frac{1}{4}^{-},f(x)\rightarrow-\infty$; as $x\rightarrow\frac{1}{4}^{+},f(x)\rightarrow+\infty$; as $x\rightarrow-\frac{2}{3}^{-},f(x)\rightarrow+\infty$; as $x\rightarrow-\frac{2}{3}^{+},f(x)\rightarrow-\infty$

Answer:

  1. $f(x)=\frac{1}{x^{2}+4x - 21}$: Vertical asymptotes $x=-7,x = 3$, Horizontal asymptote $y = 0$
  2. $f(x)=\frac{1}{x^{2}-1}$: Vertical asymptotes $x=-1,x = 1$, Horizontal asymptote $y = 0$
  3. $f(x)=\frac{x}{3x^{2}+13x - 10}$: Vertical asymptotes $x=\frac{2}{3},x=-5$, Horizontal asymptote $y = 0$
  4. $f(x)=\frac{x^{2}}{12x^{2}+5x - 2}$: Vertical asymptotes $x=\frac{1}{4},x=-\frac{2}{3}$, Horizontal asymptote $y=\frac{1}{12}$