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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.