Question

a) $lim_{x
ightarrow3}f(x)=$
b) $f(3)=$
c) $lim_{x
ightarrow1}f(x)=$
d) $f(1)=$
e) $lim_{x
ightarrow2}f(x)=$
f) $lim_{x
ightarrow - 2^{+}}f(x)=$
g) $lim_{x
ightarrow - 2^{-}}f(x)=$
h) $f(-2)=$

Explanation:

Step1: Analyze limit as x approaches 3

As \(x\) approaches 3 from both the left - hand and right - hand sides, the \(y\) - value of the function approaches 2. So, \(\lim_{x
ightarrow3}f(x)=2\).

Step2: Find the value of \(f(3)\)

The function is not defined at \(x = 3\) (open - circle at \(x = 3\)), so \(f(3)\) is undefined.

Step3: Analyze limit as x approaches 1

As \(x\) approaches 1 from both the left - hand and right - hand sides, the \(y\) - value of the function approaches 4. So, \(\lim_{x
ightarrow1}f(x)=4\).

Step4: Find the value of \(f(1)\)

The filled - in circle at \(x = 1\) has a \(y\) - value of 2, so \(f(1)=2\).

Step5: Analyze limit as x approaches 2

As \(x\) approaches 2 from both the left - hand and right - hand sides, the \(y\) - value of the function approaches 2. So, \(\lim_{x
ightarrow2}f(x)=2\).

Step6: Analyze right - hand limit as x approaches - 2

As \(x\) approaches \(-2\) from the right - hand side (\(x
ightarrow - 2^{+}\)), the \(y\) - value of the function approaches 2. So, \(\lim_{x
ightarrow - 2^{+}}f(x)=2\).

Step7: Analyze left - hand limit as x approaches - 2

As \(x\) approaches \(-2\) from the left - hand side (\(x
ightarrow - 2^{-}\)), the \(y\) - value of the function approaches 2. So, \(\lim_{x
ightarrow - 2^{-}}f(x)=2\).

Step8: Find the value of \(f(-2)\)

The filled - in circle at \(x=-2\) has a \(y\) - value of \(-2\), so \(f(-2)=-2\).

Answer:

a) \(\lim_{x
ightarrow3}f(x)=2\)
b) \(f(3)\) is undefined
c) \(\lim_{x
ightarrow1}f(x)=4\)
d) \(f(1)=2\)
e) \(\lim_{x
ightarrow2}f(x)=2\)
f) \(\lim_{x
ightarrow - 2^{+}}f(x)=2\)
g) \(\lim_{x
ightarrow - 2^{-}}f(x)=2\)
h) \(f(-2)=-2\)