Question

use the graph to find the following limits and function value for each value of a.
i) $lim_{x
ightarrow a^{-}}f(x)$ ii) $lim_{x
ightarrow a^{+}}f(x)$ iii) $lim_{x
ightarrow a}f(x)$ iv) $f(a)$, if it exists.
a. for $a = 1$ b. for $a = 3$
a. i) select the correct choice below and, if necessary, fill in the answer box within your choice.
a. $lim_{x
ightarrow1^{-}}f(x)=square$ (type an integer or a decimal.)
b. the limit does not exist.
ii) select the correct choice below and, if necessary, fill in the answer box within your choice.
a. $lim_{x
ightarrow1^{+}}f(x)=square$ (type an integer or a decimal.)
b. the limit does not exist.
iii) select the correct choice below and, if necessary, fill in the answer box within your choice.
a. $lim_{x
ightarrow1}f(x)=square$ (type an integer or a decimal.)
b. the limit does not exist.
iv) find $f(1)$. choose the correct answer below.

Explanation:

Step1: Analyze left - hand limit as x→1⁻

As x approaches 1 from the left side (x→1⁻), we look at the part of the graph to the left of x = 1. Following the curve, the y - value approaches 2. So, $\lim_{x
ightarrow1^{-}}f(x)=2$.

Step2: Analyze right - hand limit as x→1⁺

As x approaches 1 from the right side (x→1⁺), we look at the part of the graph to the right of x = 1. Following the curve, the y - value approaches 2. So, $\lim_{x
ightarrow1^{+}}f(x)=2$.

Step3: Analyze overall limit as x→1

Since $\lim_{x
ightarrow1^{-}}f(x)=\lim_{x
ightarrow1^{+}}f(x) = 2$, then $\lim_{x
ightarrow1}f(x)=2$.

Step4: Find f(1)

Looking at the graph, the solid dot at x = 1 has a y - value of 2. So, f(1)=2.

Answer:

a. i) A. $\lim_{x
ightarrow1^{-}}f(x)=2$
ii) A. $\lim_{x
ightarrow1^{+}}f(x)=2$
iii) A. $\lim_{x
ightarrow1}f(x)=2$
iv) 2