Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 2.7 : Limits at Infinity, Part I
1. For \(f\left( x \right) = 4{x^7} - 18{x^3} + 9\) evaluate each of the following limits.
- \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\)
- \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\)
Show All Solutions Hide All Solutions
a \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\) Show SolutionTo do this all we need to do is factor out the largest power of \(x\) from the whole polynomial and then use basic limit properties along with Fact 1 from this section to evaluate the limit.
\[\begin{align*}\mathop {\lim }\limits_{x \to \, - \infty } \left( {4{x^7} - 18{x^3} + 9} \right) & = \mathop {\lim }\limits_{x \to \, - \infty } \left[ {{x^7}\left( {4 - \frac{{18}}{{{x^4}}} + \frac{9}{{{x^7}}}} \right)} \right]\\ & = \left( {\mathop {\lim }\limits_{x \to \, - \infty } {x^7}} \right)\left[ {\mathop {\lim }\limits_{x \to \, - \infty } \left( {4 - \frac{{18}}{{{x^4}}} + \frac{9}{{{x^7}}}} \right)} \right] = \left( { - \infty } \right)\left( 4 \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \infty }}\end{align*}\]b \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\) Show Solution
For this part all of the mathematical manipulations we did in the first part did not depend upon the limit itself and so don’t need to be redone here. We can pick up the problem right before we actually took the limits and then proceed.
\[\mathop {\lim }\limits_{x \to \,\infty } \left( {4{x^7} - 18{x^3} + 9} \right) = \left( {\mathop {\lim }\limits_{x \to \,\infty } {x^7}} \right)\left[ {\mathop {\lim }\limits_{x \to \,\infty } \left( {4 - \frac{{18}}{{{x^4}}} + \frac{9}{{{x^7}}}} \right)} \right] = \left( \infty \right)\left( 4 \right) = \require{bbox} \bbox[2pt,border:1px solid black]{\infty }\]