4. the graphs of two functions f(x) and g(x) are depicted below.\nuse these graphs and the limit laws to…

Explanation:

Step1: Recall limit laws

The limit laws include $\lim_{x\rightarrow a}(cf(x)) = c\lim_{x\rightarrow a}f(x)$, $\lim_{x\rightarrow a}(f(x)\pm g(x))=\lim_{x\rightarrow a}f(x)\pm\lim_{x\rightarrow a}g(x)$ and $\lim_{x\rightarrow a}(f(x)g(x))=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$ (when the individual limits exist), and $\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$ ($\lim_{x\rightarrow a}g(x)\neq0$).

Step2: Find $\lim_{x\rightarrow0}f(x)$ and $\lim_{x\rightarrow0}g(x)$ from the graphs

From the graph of $f(x)$, as $x$ approaches $0$, $\lim_{x\rightarrow0}f(x) = 2$. From the graph of $g(x)$, as $x$ approaches $0$, $\lim_{x\rightarrow0}g(x)=3$.

Step3: Calculate $\lim_{x\rightarrow0}(3f(x)-4g(x))$

Using the limit - laws: [ \begin{align*} \lim_{x\rightarrow0}(3f(x)-4g(x))&=3\lim_{x\rightarrow0}f(x)-4\lim_{x\rightarrow0}g(x)\ &=3\times2 - 4\times3\ &=6 - 12\ &=- 6 \end{align*} ]

Step4: Find $\lim_{x\rightarrow3}f(x)$ and $\lim_{x\rightarrow3}g(x)$ from the graphs

From the graph of $f(x)$, as $x$ approaches $3$ from both sides, $\lim_{x\rightarrow3}f(x)=4$. From the graph of $g(x)$, as $x$ approaches $3$ from both sides, $\lim_{x\rightarrow3}g(x)=0$.

Step5: Calculate $\lim_{x\rightarrow3}f(x)g(x)$

Using the product - limit law $\lim_{x\rightarrow3}f(x)g(x)=\lim_{x\rightarrow3}f(x)\cdot\lim_{x\rightarrow3}g(x)=4\times0 = 0$.

Step6: Find $\lim_{x\rightarrow8^{-}}f(x)$ and $\lim_{x\rightarrow8^{-}}g(x)$ from the graphs

From the graph of $f(x)$, as $x$ approaches $8$ from the left, $\lim_{x\rightarrow8^{-}}f(x)=5$. From the graph of $g(x)$, as $x$ approaches $8$ from the left, $\lim_{x\rightarrow8^{-}}g(x)=1$.

Step7: Calculate $\lim_{x\rightarrow8^{-}}2f^{2}(x)g(x)$

First, $\lim_{x\rightarrow8^{-}}f^{2}(x)=(\lim_{x\rightarrow8^{-}}f(x))^{2}=5^{2}=25$. Then, using the limit laws: [ \begin{align*} \lim_{x\rightarrow8^{-}}2f^{2}(x)g(x)&=2\lim_{x\rightarrow8^{-}}f^{2}(x)\cdot\lim_{x\rightarrow8^{-}}g(x)\ &=2\times25\times1\ &=50 \end{align*} ]

Step8: Find $\lim_{x\rightarrow2}f(x)$ and $\lim_{x\rightarrow2}g(x)$ from the graphs

From the graph of $f(x)$, as $x$ approaches $2$ from both sides, $\lim_{x\rightarrow2}f(x)=0$. From the graph of $g(x)$, as $x$ approaches $2$ from both sides, $\lim_{x\rightarrow2}g(x)=2$.

Step9: Calculate $\lim_{x\rightarrow2}\frac{f(x)}{g(x)}$

Using the quotient - limit law $\lim_{x\rightarrow2}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow2}f(x)}{\lim_{x\rightarrow2}g(x)}=\frac{0}{2}=0$.

Step10: Find $\lim_{x\rightarrow5}f(x)$ and $\lim_{x\rightarrow5}g(x)$ from the graphs

From the graph of $f(x)$, as $x$ approaches $5$, $\lim_{x\rightarrow5}f(x)=4$. From the graph of $g(x)$, as $x$ approaches $5$, $\lim_{x\rightarrow5}g(x)=2$.

Step11: Calculate $\lim_{x\rightarrow5}(f(x)-g(x))$

Using the difference - limit law $\lim_{x\rightarrow5}(f(x)-g(x))=\lim_{x\rightarrow5}f(x)-\lim_{x\rightarrow5}g(x)=4 - 2=2$.

Step12: Calculate $\lim_{x\rightarrow3}(f(x)+g(x))$

We know $\lim_{x\rightarrow3}f(x)=4$ and $\lim_{x\rightarrow3}g(x)=0$. Using the sum - limit law $\lim_{x\rightarrow3}(f(x)+g(x))=\lim_{x\rightarrow3}f(x)+\lim_{x\rightarrow3}g(x)=4 + 0=4$.

Answer:

i) $-6$ ii) $0$ iii) $50$ iv) $0$ v) $2$ vi) $4$