Question

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
b. as x approaches 1 from the right, f(x)+g(x) approaches
c. f(1)+g(1)=

Explanation:

Step1: Find left - hand limit of f(x) at x = 1

From the graph of f(x), as x approaches 1 from the left, $\lim_{x
ightarrow1^{-}}f(x)=2$. From the graph of g(x), as x approaches 1 from the left, $\lim_{x
ightarrow1^{-}}g(x)=1$.

Step2: Calculate left - hand limit of f(x)+g(x)

By the sum rule of limits $\lim_{x
ightarrow a}[f(x)+g(x)]=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$. So $\lim_{x
ightarrow1^{-}}[f(x)+g(x)]=\lim_{x
ightarrow1^{-}}f(x)+\lim_{x
ightarrow1^{-}}g(x)=2 + 1=3$.

Step3: Find right - hand limit of f(x) at x = 1

From the graph of f(x), as x approaches 1 from the right, $\lim_{x
ightarrow1^{+}}f(x)=2$. From the graph of g(x), as x approaches 1 from the right, $\lim_{x
ightarrow1^{+}}g(x)= - 1$.

Step4: Calculate right - hand limit of f(x)+g(x)

Using the sum rule of limits, $\lim_{x
ightarrow1^{+}}[f(x)+g(x)]=\lim_{x
ightarrow1^{+}}f(x)+\lim_{x
ightarrow1^{+}}g(x)=2+( - 1)=1$.

Step5: Evaluate f(1)+g(1)

f(1) is undefined as there is a hole in the graph of f(x) at x = 1 and g(1) is undefined as there is a hole in the graph of g(x) at x = 1. So $f(1)+g(1)$ DNE.

Answer:

a. 3
b. 1
c. DNE