Question

the graph below is the function y = f(x)
y = f(x)
f(3)=____
lim(x→2) f(x)=____
lim(x→4) f(x)=____
lim(x→4+) f(x)=____
lim(x→4 -) f(x)=____
lim(x→1) f(x + 2)=____

Explanation:

Step1: Find f(3)

From the graph, when \(x = 3\), the solid - dot value of the function is \(1\), so \(f(3)=1\).

Step2: Find \(\lim_{x

ightarrow2}f(x)\)
As \(x\) approaches \(2\) from both the left - hand side and the right - hand side, the function values approach \(3\). So \(\lim_{x
ightarrow2}f(x)=3\).

Step3: Find \(\lim_{x

ightarrow4}f(x)\)
The left - hand limit \(\lim_{x
ightarrow4^{-}}f(x)=2\) and the right - hand limit \(\lim_{x
ightarrow4^{+}}f(x)=3\). Since the left - hand limit and the right - hand limit are not equal, \(\lim_{x
ightarrow4}f(x)\) does not exist.

Step4: Find \(\lim_{x

ightarrow4^{+}}f(x)\)
As \(x\) approaches \(4\) from the right - hand side, the function values approach \(3\). So \(\lim_{x
ightarrow4^{+}}f(x)=3\).

Step5: Find \(\lim_{x

ightarrow4^{-}}f(x)\)
As \(x\) approaches \(4\) from the left - hand side, the function values approach \(2\). So \(\lim_{x
ightarrow4^{-}}f(x)=2\).

Step6: Find \(\lim_{x

ightarrow1}f(x + 2)\)
Let \(t=x + 2\). When \(x
ightarrow1\), \(t
ightarrow3\). Then \(\lim_{x
ightarrow1}f(x + 2)=\lim_{t
ightarrow3}f(t)\). From the graph, \(\lim_{t
ightarrow3}f(t)=1\).

Answer:

\(f(3)=1\)
\(\lim_{x
ightarrow2}f(x)=3\)
\(\lim_{x
ightarrow4}f(x)\) does not exist
\(\lim_{x
ightarrow4^{+}}f(x)=3\)
\(\lim_{x
ightarrow4^{-}}f(x)=2\)
\(\lim_{x
ightarrow1}f(x + 2)=1\)