Question

evaluate the following limits. consider looking at the graph of the function to help you understand the situation. as necessary, enter oo for ∞ and -oo for −∞.
(a)
lim(x→9/6+)(21x/(9 - 6x))=
-∞
(b)
lim(x→9/6-)(21x/(9 - 6x))=

Explanation:

Step1: Analyze the limit as $x\to\frac{3}{2}^{-}$

We have the function $f(x)=\frac{21x}{9 - 6x}$. As $x\to\frac{3}{2}^{-}$, the numerator $21x\to21\times\frac{3}{2}=\frac{63}{2}>0$, and the denominator $9 - 6x\to0^{+}$ since when $x$ approaches $\frac{3}{2}$ from the left - hand side, $9-6x$ is a small positive number.

Step2: Determine the value of the limit

Using the rule $\lim_{x\to a}\frac{c}{g(x)}=\infty$ when $c>0$ and $\lim_{x\to a}g(x)=0^{+}$ ($c$ is a non - zero constant), we get $\lim_{x\to\frac{3}{2}^{-}}\frac{21x}{9 - 6x}=\infty$.

Answer:

$\infty$