Question

1. the graphs of two functions f(x) and g(x) are depicted below.

use these graphs and the limit laws to determine the values of the following limits or explain why they do not exist. justify your answers.
see example 1.5.9 in chads calculus notes for a similar type of problem
i) $lim_{x
ightarrow0}(3f(x)-4g(x))$
ii) $lim_{x
ightarrow3}f(x)g(x)$
iii) $lim_{x
ightarrow8^{-}}2f^{2}(x)g(x)$
iv) $lim_{x
ightarrow2}\frac{f(x)}{g(x)}$
v) $lim_{x
ightarrow3}(f(x)-g(x))$
vi) $lim_{x
ightarrow3}(f(x)+g(x))$

Explanation:

Step1: Recall limit laws

The limit laws include $\lim_{x
ightarrow a}(cf(x)) = c\lim_{x
ightarrow a}f(x)$, $\lim_{x
ightarrow a}(f(x)\pm g(x))=\lim_{x
ightarrow a}f(x)\pm\lim_{x
ightarrow a}g(x)$ and $\lim_{x
ightarrow a}(f(x)g(x))=\lim_{x
ightarrow a}f(x)\cdot\lim_{x
ightarrow a}g(x)$ (when the individual limits exist), and $\lim_{x
ightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow a}f(x)}{\lim_{x
ightarrow a}g(x)}$ ($\lim_{x
ightarrow a}g(x)
eq0$).

Step2: Find $\lim_{x

ightarrow0}f(x)$ and $\lim_{x
ightarrow0}g(x)$ from the graphs
From the graph of $f(x)$, as $x$ approaches $0$, $\lim_{x
ightarrow0}f(x) = 2$. From the graph of $g(x)$, as $x$ approaches $0$, $\lim_{x
ightarrow0}g(x)=3$.

Step3: Calculate $\lim_{x

ightarrow0}(3f(x)-4g(x))$
Using the limit - laws:
\[

$$\begin{align*} \lim_{x ightarrow0}(3f(x)-4g(x))&=3\lim_{x ightarrow0}f(x)-4\lim_{x ightarrow0}g(x)\\ &=3\times2 - 4\times3\\ &=6 - 12\\ &=- 6 \end{align*}$$

\]

Step4: Find $\lim_{x

ightarrow3}f(x)$ and $\lim_{x
ightarrow3}g(x)$ from the graphs
From the graph of $f(x)$, as $x$ approaches $3$ from both sides, $\lim_{x
ightarrow3}f(x)=4$. From the graph of $g(x)$, as $x$ approaches $3$ from both sides, $\lim_{x
ightarrow3}g(x)=0$.

Step5: Calculate $\lim_{x

ightarrow3}f(x)g(x)$
Using the product - limit law $\lim_{x
ightarrow3}f(x)g(x)=\lim_{x
ightarrow3}f(x)\cdot\lim_{x
ightarrow3}g(x)=4\times0 = 0$.

Step6: Find $\lim_{x

ightarrow8^{-}}f(x)$ and $\lim_{x
ightarrow8^{-}}g(x)$ from the graphs
From the graph of $f(x)$, as $x$ approaches $8$ from the left, $\lim_{x
ightarrow8^{-}}f(x)=5$. From the graph of $g(x)$, as $x$ approaches $8$ from the left, $\lim_{x
ightarrow8^{-}}g(x)=1$.

Step7: Calculate $\lim_{x

ightarrow8^{-}}2f^{2}(x)g(x)$
First, $\lim_{x
ightarrow8^{-}}f^{2}(x)=(\lim_{x
ightarrow8^{-}}f(x))^{2}=5^{2}=25$. Then, using the limit laws:
\[

$$\begin{align*} \lim_{x ightarrow8^{-}}2f^{2}(x)g(x)&=2\lim_{x ightarrow8^{-}}f^{2}(x)\cdot\lim_{x ightarrow8^{-}}g(x)\\ &=2\times25\times1\\ &=50 \end{align*}$$

\]

Step8: Find $\lim_{x

ightarrow2}f(x)$ and $\lim_{x
ightarrow2}g(x)$ from the graphs
From the graph of $f(x)$, as $x$ approaches $2$ from both sides, $\lim_{x
ightarrow2}f(x)=0$. From the graph of $g(x)$, as $x$ approaches $2$ from both sides, $\lim_{x
ightarrow2}g(x)=2$.

Step9: Calculate $\lim_{x

ightarrow2}\frac{f(x)}{g(x)}$
Using the quotient - limit law $\lim_{x
ightarrow2}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow2}f(x)}{\lim_{x
ightarrow2}g(x)}=\frac{0}{2}=0$.

Step10: Find $\lim_{x

ightarrow5}f(x)$ and $\lim_{x
ightarrow5}g(x)$ from the graphs
From the graph of $f(x)$, as $x$ approaches $5$, $\lim_{x
ightarrow5}f(x)=4$. From the graph of $g(x)$, as $x$ approaches $5$, $\lim_{x
ightarrow5}g(x)=2$.

Step11: Calculate $\lim_{x

ightarrow5}(f(x)-g(x))$
Using the difference - limit law $\lim_{x
ightarrow5}(f(x)-g(x))=\lim_{x
ightarrow5}f(x)-\lim_{x
ightarrow5}g(x)=4 - 2=2$.

Step12: Calculate $\lim_{x

ightarrow3}(f(x)+g(x))$
We know $\lim_{x
ightarrow3}f(x)=4$ and $\lim_{x
ightarrow3}g(x)=0$. Using the sum - limit law $\lim_{x
ightarrow3}(f(x)+g(x))=\lim_{x
ightarrow3}f(x)+\lim_{x
ightarrow3}g(x)=4 + 0=4$.

Answer:

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