for each function, determine the long - run behavior. \n\\(\\frac{x^{3}+1}{x^{2}+2}\\) has select an…

for each function, determine the long - run behavior. \n\\(\\frac{x^{3}+1}{x^{2}+2}\\) has select an answer\n\\(\\frac{x^{2}+1}{x^{3}+2}\\) has select an answer\n\\(\\frac{x^{2}+1}{x^{2}+2}\\) has select an answer\nhide work entry\nshow work here by typing it or attaching a file or picture\nedit insert paragraph\nboth numerator and denominator are degree 2. leading coef the long - run behavior is their ratio.\n\\(\\lim_{x\\to\\infty}\\frac{x^{2}+1}{x^{2}+2}=1\\lim_{x\\to - \\infty}\\frac{x^{2}+2}{x^{2}+1}=1\\)\nanswer: 1\nquestion 2\n\\(x^{2}+1x^{3}+2x^{3}+2x^{2}+1\\)\nnumerator degree 2, denominator degree 3. denominator gr fraction goes to 0.\n\\(\\lim_{x\\to\\infty}\\frac{x^{2}+1}{x^{3}+2}=0\\lim_{x\\to - \\infty}\\frac{x^{3}+2}{x^{2}+1}=0\\)

for each function, determine the long - run behavior. \n\\(\\frac{x^{3}+1}{x^{2}+2}\\) has select an answer\n\\(\\frac{x^{2}+1}{x^{3}+2}\\) has select an answer\n\\(\\frac{x^{2}+1}{x^{2}+2}\\) has select an answer\nhide work entry\nshow work here by typing it or attaching a file or picture\nedit insert paragraph\nboth numerator and denominator are degree 2. leading coef the long - run behavior is their ratio.\n\\(\\lim_{x\\to\\infty}\\frac{x^{2}+1}{x^{2}+2}=1\\lim_{x\\to - \\infty}\\frac{x^{2}+2}{x^{2}+1}=1\\)\nanswer: 1\nquestion 2\n\\(x^{2}+1x^{3}+2x^{3}+2x^{2}+1\\)\nnumerator degree 2, denominator degree 3. denominator gr fraction goes to 0.\n\\(\\lim_{x\\to\\infty}\\frac{x^{2}+1}{x^{3}+2}=0\\lim_{x\\to - \\infty}\\frac{x^{3}+2}{x^{2}+1}=0\\)

Answer

Explanation:

Step1: Analyze degree of numerator and denominator for $\frac{x^{3}+1}{x^{2}+2}$

Degree of numerator is 3, degree of denominator is 2. As $x\to\pm\infty$, since degree of numerator > degree of denominator, $y\to\pm\infty$.

Step2: Analyze degree of numerator and denominator for $\frac{x^{2}+1}{x^{3}+2}$

Degree of numerator is 2, degree of denominator is 3. As $x\to\pm\infty$, since degree of numerator < degree of denominator, $\lim_{x\to\pm\infty}\frac{x^{2}+1}{x^{3}+2}=0$.

Step3: Analyze degree of numerator and denominator for $\frac{x^{2}+1}{x^{2}+2}$

Degree of numerator and denominator is 2. Leading - coefficient of numerator is 1, leading - coefficient of denominator is 1. As $x\to\pm\infty$, $\lim_{x\to\pm\infty}\frac{x^{2}+1}{x^{2}+2}=\lim_{x\to\pm\infty}\frac{1+\frac{1}{x^{2}}}{1 + \frac{2}{x^{2}}}=1$.

Answer:

For $\frac{x^{3}+1}{x^{2}+2}$, $y\to\pm\infty$ as $x\to\pm\infty$; for $\frac{x^{2}+1}{x^{3}+2}$, $y\to0$ as $x\to\pm\infty$; for $\frac{x^{2}+1}{x^{2}+2}$, $y\to1$ as $x\to\pm\infty$