Paul's Online Notes
Home / Calculus I / Limits / Limit Properties
Show Mobile Notice Show All Notes Hide All Notes
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 2.4 : Limit Properties

6. Use the limit properties given in this section to compute the following limit. At each step clearly indicate the property being used. If it is not possible to compute any of the limits clearly explain why not.

$\mathop {\lim }\limits_{w \to 3} \frac{{{w^2} - 8w}}{{4 - 7w}}$
Hint : All we need to do is use the limit properties on the limit until we can use Properties 7, 8 and/or 9 from this section to compute the limit.
Show Solution
\begin{alignat*}{3}\mathop {\lim }\limits_{w \to 3} \frac{{{w^2} - 8w}}{{4 - 7w}} & = \frac{{\mathop {\lim }\limits_{w \to 3} \left( {{w^2} - 8w} \right)}}{{\mathop {\lim }\limits_{w \to 3} \left( {4 - 7w} \right)}} & & \hspace{0.25in}{\mbox{Property 4}}\\ & = \frac{{\mathop {\lim }\limits_{w \to 3} {w^2} - \mathop {\lim }\limits_{w \to 3} 8w}}{{\mathop {\lim }\limits_{w \to 3} 4 - \mathop {\lim }\limits_{w \to 3} 7w}} & & \hspace{0.25in}{\mbox{Property 2}}\\ & = \frac{{\mathop {\lim }\limits_{w \to 3} {w^2} - 8\mathop {\lim }\limits_{w \to 3} w}}{{\mathop {\lim }\limits_{w \to 3} 4 - 7\mathop {\lim }\limits_{w \to 3} w}} & & \hspace{0.25in}{\mbox{Property 1}}\\ & = \frac{{{3^2} - 8\left( 3 \right)}}{{4 - 7\left( 3 \right)}} & & \hspace{0.25in}{\mbox{Properties 7, 8, & 9}}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{15}}{{17}}}} & & & \end{alignat*}

Note that we were able to use property 4 in the first step because after evaluating the limit in the denominator we found that it wasn’t zero.