Question

for each function, determine the long run behavior.
\\( \frac { x ^ { 3 } + 1 } { x ^ { 2 } + 2 } \\) has select an answer
\\( \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 2 } \\) has select an answer
\\( \frac { x ^ { 2 } + 1 } { x ^ { 3 } + 2 } \\) has select an answer
question help: video

Explanation:

Step1: Divide numerator and denominator by highest - power of \(x\) in denominator

For \(y = \frac{x^{3}+1}{x^{2}+2}\), divide numerator and denominator by \(x^{2}\):

$$y=\frac{\frac{x^{3}}{x^{2}}+\frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}+\frac{2}{x^{2}}}=\frac{x+\frac{1}{x^{2}}}{1 + \frac{2}{x^{2}}}$$

As \(x
ightarrow\pm\infty\), \(\frac{1}{x^{2}}
ightarrow0\) and \(\frac{2}{x^{2}}
ightarrow0\). So \(y\approx x\) as \(x
ightarrow\pm\infty\)

For \(y=\frac{x^{2}+1}{x^{2}+2}\), divide numerator and denominator by \(x^{2}\):

$$y = \frac{\frac{x^{2}}{x^{2}}+\frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}+\frac{2}{x^{2}}}=\frac{1+\frac{1}{x^{2}}}{1+\frac{2}{x^{2}}}$$

As \(x
ightarrow\pm\infty\), \(\frac{1}{x^{2}}
ightarrow0\) and \(\frac{2}{x^{2}}
ightarrow0\). So \(y
ightarrow1\) as \(x
ightarrow\pm\infty\)

For \(y=\frac{x^{2}+1}{x^{3}+2}\), divide numerator and denominator by \(x^{3}\):

$$y=\frac{\frac{x^{2}}{x^{3}}+\frac{1}{x^{3}}}{\frac{x^{3}}{x^{3}}+\frac{2}{x^{3}}}=\frac{\frac{1}{x}+\frac{1}{x^{3}}}{1+\frac{2}{x^{3}}}$$

As \(x
ightarrow\pm\infty\), \(\frac{1}{x}
ightarrow0\), \(\frac{1}{x^{3}}
ightarrow0\) and \(\frac{2}{x^{3}}
ightarrow0\). So \(y
ightarrow0\) as \(x
ightarrow\pm\infty\)

Answer:

\(\frac{x^{3}+1}{x^{2}+2}\) behaves like \(y = x\) for large \(|x|\); \(\frac{x^{2}+1}{x^{2}+2}\) approaches \(y = 1\) for large \(|x|\); \(\frac{x^{2}+1}{x^{3}+2}\) approaches \(y = 0\) for large \(|x|\)