Question

ww2: problem 5
(1 point)
use the given graphs of the function ( f ) (in blue) and ( g ) (in red) to find the following limits:

1. ( lim_{x

ightarrow - 1}f(x)+g(x) )

1. ( lim_{x

ightarrow1}f(x)-g(x) )

1. ( lim_{x

ightarrow - 2}f(x)g(x) )

1. ( lim_{x

ightarrow0}\frac{f(x)}{g(x)} ) (type dne if the limit does not exist)

1. ( lim_{x

ightarrow - 2}sqrt{3 - f(x)} )
note: type inf for ( infty ) and -inf for ( -infty ). if the limit does not exist, write dne.
you can click on the graph to enlarge the image.
note: in order to get credit for this problem all answers must be correct.

Explanation:

Step1: Recall limit - sum rule

$\lim_{x
ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$

Step2: Recall limit - product rule

$\lim_{x
ightarrow a}(f(x)g(x))=\lim_{x
ightarrow a}f(x)\cdot\lim_{x
ightarrow a}g(x)$

Step3: Recall limit - quotient rule

$\lim_{x
ightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow a}f(x)}{\lim_{x
ightarrow a}g(x)}$, $\lim_{x
ightarrow a}g(x)
eq0$

Step4: Recall limit - root rule

$\lim_{x
ightarrow a}\sqrt[n]{h(x)}=\sqrt[n]{\lim_{x
ightarrow a}h(x)}$ for appropriate $n$ and $\lim_{x
ightarrow a}h(x)\geq0$ when $n$ is even.

Step5: Analyze the graphs

To find the limits, we look at the behavior of the blue - colored graph of $y = f(x)$ and the red - colored graph of $y = g(x)$ as $x$ approaches the relevant values. For example, to find $\lim_{x
ightarrow a}(f(x)+g(x))$, we find $\lim_{x
ightarrow a}f(x)$ by looking at the $y$ - value that $f(x)$ approaches as $x
ightarrow a$ on the blue graph and $\lim_{x
ightarrow a}g(x)$ by looking at the $y$ - value that $g(x)$ approaches as $x
ightarrow a$ on the red graph, and then add them.

Since the specific values of $x$ to which we are taking the limits are not clearly visible in the provided image, assume we are taking limits as $x
ightarrow x_0$. Let $\lim_{x
ightarrow x_0}f(x)=L_1$ and $\lim_{x
ightarrow x_0}g(x)=L_2$.

1. $\lim_{x

ightarrow x_0}(f(x)+g(x))=L_1 + L_2$

1. $\lim_{x

ightarrow x_0}(f(x)-g(x))=L_1 - L_2$

1. $\lim_{x

ightarrow x_0}(f(x)g(x))=L_1\times L_2$

1. $\lim_{x

ightarrow x_0}\frac{f(x)}{g(x)}=\frac{L_1}{L_2}$ (if $L_2
eq0$)

1. $\lim_{x

ightarrow x_0}\sqrt{3 - f(x)}=\sqrt{3 - L_1}$ (if $3 - L_1\geq0$)

However, without the exact values of the limits of $f(x)$ and $g(x)$ from the graph for a specific $x$ - value, we cannot give numerical answers. If we assume we are taking limits as $x
ightarrow1$ (for example), we would estimate the $y$ - values of $f(x)$ and $g(x)$ as $x$ approaches 1 from the left and right and use the above rules.

Answer:

Since the specific $x$ - values for which the limits are taken are not clear from the image, we cannot provide numerical answers. The general rules for calculating the limits are:

1. $\lim_{x

ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$

1. $\lim_{x

ightarrow a}(f(x)-g(x))=\lim_{x
ightarrow a}f(x)-\lim_{x
ightarrow a}g(x)$

1. $\lim_{x

ightarrow a}(f(x)g(x))=\lim_{x
ightarrow a}f(x)\cdot\lim_{x
ightarrow a}g(x)$

1. $\lim_{x

ightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow a}f(x)}{\lim_{x
ightarrow a}g(x)}$ ($\lim_{x
ightarrow a}g(x)
eq0$)

1. $\lim_{x

ightarrow a}\sqrt{3 - f(x)}=\sqrt{3-\lim_{x
ightarrow a}f(x)}$ ($3-\lim_{x
ightarrow a}f(x)\geq0$)