Question

evaluating limits in exercises 37 - 40, use the information to evaluate the limits.

1. $lim_{x

ightarrow c}f(x)=\frac{2}{5}$
$lim_{x
ightarrow c}g(x)=2$
(a) $lim_{x
ightarrow c}5g(x)$
(b) $lim_{x
ightarrow c}f(x)+g(x)$
(c) $lim_{x
ightarrow c}f(x)g(x)$
(d) $lim_{x
ightarrow c}\frac{f(x)}{g(x)}$

1. $lim_{x

ightarrow c}f(x)=2$
$lim_{x
ightarrow c}g(x)=\frac{3}{4}$
(a) $lim_{x
ightarrow c}4f(x)$
(b) $lim_{x
ightarrow c}f(x)+g(x)$
(c) $lim_{x
ightarrow c}f(x)g(x)$
(d) $lim_{x
ightarrow c}\frac{f(x)}{g(x)}$

Explanation:

Step1: Recall limit - constant multiple rule

For $\lim_{x
ightarrow c}[5g(x)]$, by the constant - multiple rule of limits $\lim_{x
ightarrow c}[k\cdot h(x)]=k\cdot\lim_{x
ightarrow c}h(x)$, where $k = 5$ and $h(x)=g(x)$. Given $\lim_{x
ightarrow c}g(x)=2$, then $\lim_{x
ightarrow c}[5g(x)]=5\times\lim_{x
ightarrow c}g(x)=5\times2 = 10$.

Step2: Recall limit - sum rule

For $\lim_{x
ightarrow c}[f(x)+g(x)]$, by the sum rule of limits $\lim_{x
ightarrow c}[h(x)+k(x)]=\lim_{x
ightarrow c}h(x)+\lim_{x
ightarrow c}k(x)$. Given $\lim_{x
ightarrow c}f(x)=\frac{2}{5}$ and $\lim_{x
ightarrow c}g(x)=2$, then $\lim_{x
ightarrow c}[f(x)+g(x)]=\lim_{x
ightarrow c}f(x)+\lim_{x
ightarrow c}g(x)=\frac{2}{5}+2=\frac{2 + 10}{5}=\frac{12}{5}$.

Step3: Recall limit - product rule

For $\lim_{x
ightarrow c}[f(x)g(x)]$, by the product rule of limits $\lim_{x
ightarrow c}[h(x)\cdot k(x)]=\lim_{x
ightarrow c}h(x)\cdot\lim_{x
ightarrow c}k(x)$. Given $\lim_{x
ightarrow c}f(x)=\frac{2}{5}$ and $\lim_{x
ightarrow c}g(x)=2$, then $\lim_{x
ightarrow c}[f(x)g(x)]=\lim_{x
ightarrow c}f(x)\cdot\lim_{x
ightarrow c}g(x)=\frac{2}{5}\times2=\frac{4}{5}$.

Step4: Recall limit - quotient rule

For $\lim_{x
ightarrow c}\frac{f(x)}{g(x)}$, by the quotient rule of limits $\lim_{x
ightarrow c}\frac{h(x)}{k(x)}=\frac{\lim_{x
ightarrow c}h(x)}{\lim_{x
ightarrow c}k(x)}$ ($\lim_{x
ightarrow c}k(x)
eq0$). Given $\lim_{x
ightarrow c}f(x)=\frac{2}{5}$ and $\lim_{x
ightarrow c}g(x)=2$, then $\lim_{x
ightarrow c}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow c}f(x)}{\lim_{x
ightarrow c}g(x)}=\frac{\frac{2}{5}}{2}=\frac{2}{5}\times\frac{1}{2}=\frac{1}{5}$.

For the second - set of limits:

Step1: Recall limit - constant multiple rule

For $\lim_{x
ightarrow c}[4f(x)]$, by the constant - multiple rule of limits $\lim_{x
ightarrow c}[k\cdot h(x)]=k\cdot\lim_{x
ightarrow c}h(x)$, where $k = 4$ and $h(x)=f(x)$. Given $\lim_{x
ightarrow c}f(x)=2$, then $\lim_{x
ightarrow c}[4f(x)]=4\times\lim_{x
ightarrow c}f(x)=4\times2 = 8$.

Step2: Recall limit - sum rule

For $\lim_{x
ightarrow c}[f(x)+g(x)]$, by the sum rule of limits $\lim_{x
ightarrow c}[h(x)+k(x)]=\lim_{x
ightarrow c}h(x)+\lim_{x
ightarrow c}k(x)$. Given $\lim_{x
ightarrow c}f(x)=2$ and $\lim_{x
ightarrow c}g(x)=\frac{3}{4}$, then $\lim_{x
ightarrow c}[f(x)+g(x)]=\lim_{x
ightarrow c}f(x)+\lim_{x
ightarrow c}g(x)=2+\frac{3}{4}=\frac{8 + 3}{4}=\frac{11}{4}$.

Step3: Recall limit - product rule

For $\lim_{x
ightarrow c}[f(x)g(x)]$, by the product rule of limits $\lim_{x
ightarrow c}[h(x)\cdot k(x)]=\lim_{x
ightarrow c}h(x)\cdot\lim_{x
ightarrow c}k(x)$. Given $\lim_{x
ightarrow c}f(x)=2$ and $\lim_{x
ightarrow c}g(x)=\frac{3}{4}$, then $\lim_{x
ightarrow c}[f(x)g(x)]=\lim_{x
ightarrow c}f(x)\cdot\lim_{x
ightarrow c}g(x)=2\times\frac{3}{4}=\frac{3}{2}$.

Step4: Recall limit - quotient rule

For $\lim_{x
ightarrow c}\frac{f(x)}{g(x)}$, by the quotient rule of limits $\lim_{x
ightarrow c}\frac{h(x)}{k(x)}=\frac{\lim_{x
ightarrow c}h(x)}{\lim_{x
ightarrow c}k(x)}$ ($\lim_{x
ightarrow c}k(x)
eq0$). Given $\lim_{x
ightarrow c}f(x)=2$ and $\lim_{x
ightarrow c}g(x)=\frac{3}{4}$, then $\lim_{x
ightarrow c}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow c}f(x)}{\lim_{x
ightarrow c}g(x)}=\frac{2}{\frac{3}{4}}=2\times\frac{4}{3}=\frac{8}{3}$.

Answer:

**For the first set of limits (where $\lim_{x
ightarrow c}f(x)=\frac{2}{5}$ and $\lim_{x
ightarrow c}g(x)=2$):**
(a) $10$
(b) $\frac{12}{5}$
(c) $\frac{4}{5}$
(d) $\frac{1}{5}$

**For the second set of limits (where $\lim_{x
ightarrow c}f(x)=2$ and $\lim_{x
ightarrow c}g(x)=\frac{3}{4}$):**
(a) $8$
(b) $\frac{11}{4}$
(c) $\frac{3}{2}$
(d) $\frac{8}{3}$