Question

worksheet a: (topic 1.7) rational functions and end behavior name: luciana bustamante p:3 directions: for each of the following, determine if the given rational function has a horizontal asymptote. if it does, write the equation of the horizontal asymptote. 1. $f(x)=\frac{3x^{2}-1}{2x^{2}+5x + 7}$ horizontal asymptote: y or n if yes, equation: $f(x)=\frac{3}{2}$ 2. $g(x)=\frac{x^{2}+x + 4}{5x^{2}+7x + 8}$ horizontal asymptote: y or n if yes, equation: $g(x)=\frac{1}{5}$ 3. $h(x)=\frac{5x^{3}-2x^{2}-1}{x^{2}-6}$ horizontal asymptote: y or n if yes, equation: $h(x)=5x$ 4. $k(x)=\frac{6x^{2}+2x + 3}{2x^{2}-11x + 4}$ horizontal asymptote: y or n if yes, equation: 5. $r(x)=\frac{(2x - 1)(5x + 6)}{(x + 3)(x - 6)}$ horizontal asymptote: y or n if yes, equation: 6. $q(x)=\frac{(x^{2}-3)^{2}}{3x^{4}+4x^{2}+7}$ horizontal asymptote: y or n if yes, equation: 7. $p(x)=\frac{(3x - 1)^{2}}{2x^{2}+3x + 5}$ horizontal asymptote: y or n if yes, equation: 8. $y=\frac{(x - 2)(4 - x)}{(x + 3)^{2}}$ horizontal asymptote: y or n if yes, equation: 9. $s(x)=\frac{(2x^{2}+3)^{2}(x - 4)}{(x^{2}+5)(x - 2)}$ horizontal asymptote: y or n if yes, equation: directions: write limit statements for the end - behavior of the following rational functions. 10. $y=\frac{2x^{3}-5x + 6}{6x^{3}+10x^{2}-4x - 12}$ left: $-infty$ right: $infty$ 11. $y=\frac{(4x + 3)^{2}}{(3x - 1)(2x + 5)}$ left: right: 12. $y=\frac{x^{2}}{(x - 1)^{2}}$ left: right:

Explanation:

Step1: Recall the rules for horizontal asymptotes of rational functions

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, where $a_n
eq0$ and $b_m
eq0$:

1. If $n < m$, the horizontal asymptote is $y = 0$.
2. If $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$.
3. If $n>m$, there is no horizontal asymptote.

Step2: Analyze $k(x)=\frac{6x^2 + 2x+3}{2x^2-11x + 4}$

Here $n = 2$ (degree of numerator) and $m = 2$ (degree of denominator). $a_n=6$ and $b_m = 2$.
Since $n=m$, the horizontal asymptote is $y=\frac{6}{2}=3$.

Step3: Analyze $r(x)=\frac{(2x - 1)(5x+6)}{(x + 3)(x - 6)}=\frac{10x^2+7x - 6}{x^2-3x - 18}$

Here $n = 2$ (degree of numerator) and $m = 2$ (degree of denominator). $a_n = 10$ and $b_m=1$.
Since $n=m$, the horizontal asymptote is $y = 10$.

Step4: Analyze $q(x)=\frac{(x^2-3)^2}{3x^4+4x^2+7}=\frac{x^4-6x^2 + 9}{3x^4+4x^2+7}$

Here $n = 4$ (degree of numerator) and $m = 4$ (degree of denominator). $a_n=1$ and $b_m = 3$.
Since $n=m$, the horizontal asymptote is $y=\frac{1}{3}$.

Step5: Analyze $p(x)=\frac{(3x - 1)^2}{2x^2+3x+5}=\frac{9x^2-6x + 1}{2x^2+3x+5}$

Here $n = 2$ (degree of numerator) and $m = 2$ (degree of denominator). $a_n=9$ and $b_m=2$.
Since $n=m$, the horizontal asymptote is $y=\frac{9}{2}$.

Step6: Analyze $y=\frac{(x - 2)(4 - x)}{(x + 3)^2}=\frac{-x^2+6x - 8}{x^2+6x + 9}$

Here $n = 2$ (degree of numerator) and $m = 2$ (degree of denominator). $a_n=-1$ and $b_m=1$.
Since $n=m$, the horizontal asymptote is $y=-1$.

Step7: Analyze $s(x)=\frac{(2x^2+3)^2(x - 4)}{(x^2+5)(x - 2)}=\frac{(4x^4 + 12x^2+9)(x - 4)}{x^3-2x^2+5x - 10}=\frac{4x^5-16x^4+12x^3-48x^2+9x - 36}{x^3-2x^2+5x - 10}$

Here $n = 5$ (degree of numerator) and $m = 3$ (degree of denominator). Since $n>m$, there is no horizontal asymptote.

Answer:

1. Horizontal Asymptote: Y, Equation: $y = 3$
2. Horizontal Asymptote: Y, Equation: $y = 10$
3. Horizontal Asymptote: Y, Equation: $y=\frac{1}{3}$
4. Horizontal Asymptote: Y, Equation: $y=\frac{9}{2}$
5. Horizontal Asymptote: Y, Equation: $y=-1$
6. Horizontal Asymptote: N