Question

use the graph to find the following limits and function value
for each value of a.
i) $\lim\limits_{x\to a^-} f(x)$ ii) $\lim\limits_{x\to a^+} f(x)$ iii) $\lim\limits_{x\to a} f(x)$ iv) $f(a)$, if it exists.
a. for $a = 1$ b. for $a = - 1$
ii) select the correct choice below and, if necessary, fill in the answer box within your choice.
a. $\lim\limits_{x\to 1^+} f(x) = 2$ (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. $\lim\limits_{x\to 1} f(x) = \square$ (type an integer or a decimal.)
b. the limit does not exist.
iv) find $f(1)$. choose the correct answer below.
a. 1
b. $-2$
c. 3
d. the answer is undefined.

Explanation:

ii) For $\boldsymbol{\lim_{x \to 1^+} f(x)}$

Step1: Analyze the right - hand limit

To find the right - hand limit as $x$ approaches $1$ (i.e., $\lim_{x \to 1^+} f(x)$), we look at the behavior of the function as $x$ gets closer to $1$ from values greater than $1$. From the graph, when we move along the part of the graph that is to the right of $x = 1$ and approach $x = 1$, the $y$ - value that the function approaches is $2$. So, $\lim_{x \to 1^+} f(x)=2$.

Step1: Recall the definition of the limit

The limit $\lim_{x \to a} f(x)$ exists if and only if $\lim_{x \to a^-} f(x)=\lim_{x \to a^+} f(x)$.

Step2: Analyze left - hand and right - hand limits

We already know that $\lim_{x \to 1^+} f(x) = 2$. Now, we need to find $\lim_{x \to 1^-} f(x)$. From the graph, when we approach $x = 1$ from the left (values less than $1$), the function approaches a different $y$ - value (from the left - hand side of the graph at $x = 1$, the $y$ - value is $5$ or some other value different from $2$). Since $\lim_{x \to 1^-} f(x)
eq\lim_{x \to 1^+} f(x)$, the limit $\lim_{x \to 1} f(x)$ does not exist.

Step1: Analyze the function value at $x = 1$

To find $f(1)$, we look at the point on the graph where $x = 1$. The solid dot (which represents the function value) at $x = 1$ has a $y$ - value of $3$. So, $f(1)=3$.

Answer:

$\lim_{x \to 1^+} f(x)=\boldsymbol{2}$ (corresponding to option A)

iii) For $\boldsymbol{\lim_{x \to 1} f(x)}$