Question

precalculus 2.2 ws limits of polynomials
name: rosene jenny mistok
use the following functions to evaluate the limits. sketch a graph to help if needed.
$f(x)=x^{3}-2x^{2}+4x$
$g(x)=-3x^{4}+2x$

1. $limlimits_{x\to-infty}f(x)=$
2. $limlimits_{x\toinfty}g(x)=$
3. $limlimits_{x\toinfty}f(x)=$

evaluate the limit

1. $limlimits_{x\to-infty}5x^{2}-3x + 1=$
2. $limlimits_{x\toinfty}-x^{5}+9=$
3. $limlimits_{x\to-infty}2x - 3=$

Explanation:

Step1: Analyze the leading term of \(f(x)=x^{3}-2x^{2}+4x\)

For \(x\to-\infty\), the leading term is \(x^{3}\). As \(x\to-\infty\), \(x^{3}\to-\infty\). So \(\lim_{x\to-\infty}f(x)=-\infty\)

Step2: Analyze the leading term of \(g(x)=-3x^{4}+2x\)

For \(x\to\infty\), the leading term is \(-3x^{4}\). As \(x\to\infty\), \(x^{4}\to\infty\) and \(-3x^{4}\to-\infty\). So \(\lim_{x\to\infty}g(x)=-\infty\)

Step3: Analyze the leading term of \(f(x)=x^{3}-2x^{2}+4x\)

For \(x\to\infty\), the leading term is \(x^{3}\). As \(x\to\infty\), \(x^{3}\to\infty\). So \(\lim_{x\to\infty}f(x)=\infty\)

Step4: Analyze the leading term of \(y = 5x^{2}-3x + 1\)

For \(x\to-\infty\), the leading term is \(5x^{2}\). As \(x\to-\infty\), \(x^{2}\to\infty\) and \(5x^{2}\to\infty\). So \(\lim_{x\to-\infty}(5x^{2}-3x + 1)=\infty\)

Step5: Analyze the leading term of \(y=-x^{5}+9\)

For \(x\to\infty\), the leading term is \(-x^{5}\). As \(x\to\infty\), \(x^{5}\to\infty\) and \(-x^{5}\to-\infty\). So \(\lim_{x\to\infty}(-x^{5}+9)=-\infty\)

Step6: Analyze the function \(y = 2x-3\)

For \(x\to-\infty\), as \(x\to-\infty\), \(2x\to-\infty\). So \(\lim_{x\to-\infty}(2x - 3)=-\infty\)

Answer:

1. \(-\infty\)
2. \(-\infty\)
3. \(\infty\)
4. \(\infty\)
5. \(-\infty\)
6. \(-\infty\)