91Ó°ÊÓ

A function f dominates another function g as $x→∞$if $f\left(x\right),g\left(x\right)$both grow without bound as $x→∞$and if $\underset{x→∞}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=∞$.

Intuitively, f dominates gas $x→∞$if $f\left(x\right)$is very much larger than $g\left(x\right)$for very large values of x. Use limits to determine whether $u\left(x\right)$dominates $v\left(x\right)$or $v\left(x\right)$dominates $u\left(x\right)$or neither.

$$\begin{gathered} u\left(x\right)=0.001x^2-100x, \\ v\left(x\right)=100{\log }_{3}x \end{gathered}$$

Sketch careful, labeled graphs of each function $f$in Exercises $57-82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\frac{x^2\left(x-1\right)}{{\left(x-2\right)}^2}$.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=\frac{x^2-x}{x^2-3x+2}$

A function f dominates another function g as $x→∞$if $f\left(x\right),g\left(x\right)$both grow without bound as $x→∞$and if $\underset{x→∞}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=∞$.

Intuitively, f dominates gas $x→∞$if $f\left(x\right)$is very much larger than $g\left(x\right)$for very large values of x. Use limits to determine whether $u\left(x\right)$dominates $v\left(x\right)$or $v\left(x\right)$dominates $u\left(x\right)$or neither.

$$\begin{gathered} u\left(x\right)=0.001x^2-100x, \\ v\left(x\right)=100{\log }_{3}x \end{gathered}$$

Sketch careful, labeled graphs of each function $f$in Exercises $57-82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\frac{x^3}{x^2-3x+2}$.

As you will prove in Exercises 93 and 94, exponential growth functions $e^{kx}$always dominate power functions $x^r$, and power functions $x^r$with positive powers always dominate logarithmic functions ${\log }_{b}x$. Use these facts to quickly determine each of the limits in Exercises 75-80.

$\underset{x→∞}{\lim }\frac{x^{101}+500}{e^x}$

Sketch careful, labeled graphs of each function $f$in Exercises $57-82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=x\ln x$.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=\frac{1-x}{e^x}$

Sketch careful, labeled graphs of each function $f$in Exercises $57-82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\frac{2^x}{1-2^x}$.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=\ln (x^2+1)$