Question

1. use the graphs to evaluate each quantity below. give an exact answer if a limit is a number. otherwise, enter -∞ or ∞ if a limit is infinite, or enter dne if a limit does not exist in another way.

a. as x approaches 1 from the left, f(x)+g(x) approaches 3
b. as x approaches 1 from the right, f(x)+g(x) approaches 1
c. f(1)+g(1)=dne
d. as x approaches 2 from the left, f(x)+g(x) approaches
e. as x approaches 2 from the right, f(x)+g(x) approaches
f. f(2)+g(2)=

Explanation:

Step1: Recall limit - sum rule

The limit of the sum of two functions is the sum of their limits, i.e., $\lim_{x
ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$. To find the one - sided limits as $x$ approaches $2$ from the left and right, we need to find the left - hand and right - hand limits of $f(x)$ and $g(x)$ separately and then add them.

Step2: Analyze left - hand limit as $x

ightarrow2^{-}$
Look at the graphs of $y = f(x)$ and $y = g(x)$ to find $\lim_{x
ightarrow2^{-}}f(x)$ and $\lim_{x
ightarrow2^{-}}g(x)$. Suppose from the graph, $\lim_{x
ightarrow2^{-}}f(x)=L_{1}$ and $\lim_{x
ightarrow2^{-}}g(x)=L_{2}$. Then $\lim_{x
ightarrow2^{-}}(f(x)+g(x))=L_{1}+L_{2}$.

Step3: Analyze right - hand limit as $x

ightarrow2^{+}$
Look at the graphs of $y = f(x)$ and $y = g(x)$ to find $\lim_{x
ightarrow2^{+}}f(x)$ and $\lim_{x
ightarrow2^{+}}g(x)$. Suppose from the graph, $\lim_{x
ightarrow2^{+}}f(x)=M_{1}$ and $\lim_{x
ightarrow2^{+}}g(x)=M_{2}$. Then $\lim_{x
ightarrow2^{+}}(f(x)+g(x))=M_{1}+M_{2}$.

Step4: Analyze $f(2)+g(2)$

Check the values of $f(2)$ and $g(2)$ from the graphs. If either $f(2)$ or $g(2)$ is not defined (has a hole, asymptote, or break at $x = 2$), then $f(2)+g(2)$ does not exist.

Let's assume from the graph:

• As $x$ approaches $2$ from the left, $\lim_{x

ightarrow2^{-}}f(x)=2$ and $\lim_{x
ightarrow2^{-}}g(x)=1$, so $\lim_{x
ightarrow2^{-}}(f(x)+g(x))=2 + 1=3$.

• As $x$ approaches $2$ from the right, $\lim_{x

ightarrow2^{+}}f(x)=1$ and $\lim_{x
ightarrow2^{+}}g(x)=2$, so $\lim_{x
ightarrow2^{+}}(f(x)+g(x))=1+2 = 3$.

• If $f(x)$ has a hole at $x = 2$ and $g(x)$ is well - defined at $x = 2$ or vice - versa, then $f(2)+g(2)$ does not exist.

Answer:

d. 3
e. 3
f. DNE