Question

1. use the graph to find the following. write ±∞ or does not exist where appropriate.

a) $lim_{x
ightarrow3^{+}}f(x)=$
b) $lim_{x
ightarrow3^{-}}f(x)=$
c) $lim_{x
ightarrow3}f(x)=$
d) $lim_{x
ightarrow - 2^{-}}f(x)=$
e) $lim_{x
ightarrow4^{+}}f(x)=$
f) $lim_{x
ightarrow - 2}f(x)=$
g) $f(3.2)=$
h) $f(-3)=$
i) $f(-1.3)=$

Explanation:

Step1: Analyze right - hand limit as x approaches 3

As x approaches 3 from the right ($x\to3^{+}$), we look at the part of the graph for $x > 3$. The function approaches - 4. So, $\lim_{x\to3^{+}}f(x)=-4$.

Step2: Analyze left - hand limit as x approaches 3

As x approaches 3 from the left ($x\to3^{-}$), we look at the part of the graph for $x < 3$. The function approaches 2. So, $\lim_{x\to3^{-}}f(x)=2$.

Step3: Analyze overall limit as x approaches 3

Since $\lim_{x\to3^{+}}f(x)
eq\lim_{x\to3^{-}}f(x)$, $\lim_{x\to3}f(x)$ does not exist.

Step4: Analyze left - hand limit as x approaches - 2

As x approaches - 2 from the left ($x\to - 2^{-}$), the function approaches - 2. So, $\lim_{x\to - 2^{-}}f(x)=-2$.

Step5: Analyze right - hand limit as x approaches 4

As x approaches 4 from the right ($x\to4^{+}$), the function approaches $-\infty$. So, $\lim_{x\to4^{+}}f(x)=-\infty$.

Step6: Analyze overall limit as x approaches - 2

Since the function is continuous at $x=-2$, $\lim_{x\to - 2}f(x)=-2$.

Step7: Analyze derivative at x = 3.2

At $x = 3.2$, the function has a sharp corner, so the derivative $f^{\prime}(3.2)$ does not exist.

Step8: Analyze derivative at x=-3

At $x=-3$, the function is a horizontal line, so the slope (derivative) $f^{\prime}(-3)=0$.

Step9: Analyze derivative at x=-1.3

At $x=-1.3$, the function is a straight - line segment with a slope of 1, so $f^{\prime}(-1.3)=1$.

Answer:

a. $\lim_{x\to3^{+}}f(x)=-4$
b. $\lim_{x\to3^{-}}f(x)=2$
c. $\lim_{x\to3}f(x)$ does not exist
d. $\lim_{x\to - 2^{-}}f(x)=-2$
e. $\lim_{x\to4^{+}}f(x)=-\infty$
f. $\lim_{x\to - 2}f(x)=-2$
g. $f^{\prime}(3.2)$ does not exist
h. $f^{\prime}(-3)=0$
i. $f^{\prime}(-1.3)=1$