Answer
a) $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$, since, as x approaches 3 from the left, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 1.
b) $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$, since, as x approaches 3 from the right, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3.
c) $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$, does not exist, because both the left hand and right-hand limits are unequal at $ x=3$.
d) $ f\left( 3 \right)=3$, because this point $\left( 3,3 \right)$ is shown by the solid dot in the provided graph.
e) $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$, since, as x approaches 3.5 from the left, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3.
f) $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$, since, as x approaches 3.5 from the right, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3.
g) $\underset{x\to 3.5}{\mathop{\lim }}\,f\left( x \right)=3$, because the left hand and right-hand limits at $ x=3.5$ are equal.
h) $ f\left( 3.5 \right)=3$, because this point is shown by the solid dot in the provided graph.
Work Step by Step
(a)
Consider the provided limit $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$.
To find $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3$ but from the left.
As x approaches 3 from the left, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 1. This point $\left( 3,1 \right)$ is shown by the open dot in the above graph.
The point $\left( 3,1 \right)$ has a y-coordinate of 1.
Thus, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$.
Hence, the value of $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 1.
(b)
Consider the provided limit $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$.
To find $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3$ but from the right.
As x approaches 3 from the right, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3,3 \right)$ is shown by the solid dot in the above graph.
The point $\left( 3,3 \right)$ has a y-coordinate of 3.
Thus, $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Hence, the value of $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3.
(c)
Consider the provided limit $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$.
To find $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3$ but from the right.
As x approaches 3 from the right, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3,3 \right)$ is shown by the solid dot in the above graph.
The point $\left( 3,3 \right)$ has a y-coordinate of 3.
Thus, $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Hence, the value of $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3.
To find $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3$ but from the left.
As x approaches 3 from the left, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 1. This point $\left( 3,1 \right)$ is shown by the open dot in the above graph.
The point $\left( 3,1 \right)$ has a y-coordinate of 1.
Thus, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$.
Hence, the value of $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 1.
Since, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$ and $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Here, both the left-hand limit and right-hand limit at $ x=3$ are unequal,
Hence, $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$ does not exist.
(d)
Consider the provided function, $ f\left( 3 \right)$.
To find $ f\left( 3 \right)$, examine the portion of the graph near $ x=3$.
The graph of “f” at $ x=3$ is shown by the solid dot in the provided graph with coordinates $\left( 3,3 \right)$.
Thus, $ f\left( 3 \right)=3$.
(e)
Consider the provided limit $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$.
To find $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3.5$ but from the left.
As x approaches 3.5 from the left, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $ y=3$ in the above graph.
The point $\left( 3.5,3 \right)$ has a y-coordinate of 3.
Thus, $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Hence, the value of $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 3.
(f)
Consider the provided limit $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$.
To find $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3.5$ but from the right.
As x approaches 3.5 from the right, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $ y=3$ in the above graph
Thus $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Hence, the value of $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3.
(g)
Consider the provided limit $\underset{x\to 3.5}{\mathop{\lim }}\,f\left( x \right)$.
To find $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3.5$ but from the right.
As x approaches 3.5 from the right, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $ y=3$ in the above graph
Thus $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Hence, the value of $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3.
To find $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $ x=3.5$ but from the left.
As x approaches 3.5 from the left, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $ y=3$ in the above graph.
The point $\left( 3.5,3 \right)$ has a y-coordinate of 3.
Thus, $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Hence, the value of $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 3.
Since, $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ and $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$.
Here, both the left-hand limit and right-hand limit at $ x=3.5$ are equal,
Hence, $\underset{x\to 3.5}{\mathop{\lim }}\,f\left( x \right)=3$.
(h)
To find $ f\left( 3.5 \right)$, examine the portion of the graph near $ x=3.5$.
The graph of “f” at $ x=3.5$ is shown by the solid dot in the provided graph with coordinates $\left( 3.5,3 \right)$.
Thus, the function $ f\left( 3.5 \right)=3$.
