Chapter 4.1, Problem 13AYU

If $f\left(x\right)=-2x^5+x^3-5x^2+7$, then $\underset{x\to -\infty }{\lim }f\left(x\right)=$ _____ and $\underset{x\to \infty }{\lim }f\left(x\right)=$ _____.

To determine

To fill: For the function $f\left(x\right)=-2x^5+x^3-5x^2+7$, ${\underset{}{\lim }}_{x\to -\infty }f\left(x\right)={}_{\_\_\_\_\_}$ and $\underset{}{{\lim }_{x\to \infty }}f\left(x\right)={}_{\_\_\_\_\_}$.

Answer to Problem 13AYU

$\underset{\_}{\text{For the function }f\left(x\right)=-2x^5+x^3-5x^2+7,{\underset{}{\lim }}_{x\to -\infty }f\left(x\right)=\infty \text{ and}{\underset{}{\lim }}_{x\to \infty }f\left(x\right)=-\infty {}_{\_\_\_\_\_}}$.

Explanation of Solution

Given:

$f\left(x\right)=-2x^5+x^3-5x^2+7$

$y=f\left(x\right)=a_n{x}^n$

Here $a_n=-2<0,n=5$.

If $a_n<0$, for increasing values of $x$ approaching to positive infinity value of limit of $y=f\left(x\right)$ decreases in the negative direction equals negative infinity.

If $a_n<0$, for decreasing values of $x$ approaching to negative infinity value of limit of $y=f\left(x\right)$ increases in the positive direction equals positive infinity.

${\underset{}{\lim }}_{x\to -\infty }f\left(x\right)=\underset{x\to -\infty }{\text{lim}}-2x^5+x^3-5x^2+7=\infty$

$\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\text{lim}}-2x^5+x^3-5x^2+7=-\infty$