Question

1. (5 points) the graph of y = h(x) is shown below. for each question, write a specific number, ∞, -∞, or dne.

(a) $lim_{x
ightarrow - 1}h(x)=$(b) $lim_{x
ightarrow - 1^{-}}h(x)=$(c) $lim_{x
ightarrow - 1^{+}}h(x)=$
(d) $lim_{x
ightarrow1}h(x)=$(e) $lim_{x
ightarrow1^{-}}h(x)=$(f) $lim_{x
ightarrow1^{+}}h(x)=$
(g) $lim_{x
ightarrow4}h(x)=$(h) $lim_{x
ightarrow4^{-}}h(x)=$(i) $lim_{x
ightarrow4^{+}}h(x)=$
(j) $h(1)=$

Explanation:

Step1: Analyze limit as $x\to - 1$

Approaching $x = - 1$ from both left and right, the function value approaches $1$. So $\lim_{x\to - 1}h(x)=1$.

Step2: Analyze left - hand limit as $x\to - 1$

Approaching $x=-1$ from the left side, the function value approaches $1$. So $\lim_{x\to - 1^{-}}h(x)=1$.

Step3: Analyze right - hand limit as $x\to - 1$

Approaching $x = - 1$ from the right side, the function value approaches $1$. So $\lim_{x\to - 1^{+}}h(x)=1$.

Step4: Analyze limit as $x\to1$

The left - hand limit and right - hand limit as $x\to1$ are different. $\lim_{x\to1^{-}}h(x)=1$ and $\lim_{x\to1^{+}}h(x)=2$. Since they are not equal, $\lim_{x\to1}h(x)$ DNE.

Step5: Analyze left - hand limit as $x\to1$

Approaching $x = 1$ from the left side, the function value approaches $1$. So $\lim_{x\to1^{-}}h(x)=1$.

Step6: Analyze right - hand limit as $x\to1$

Approaching $x = 1$ from the right side, the function value approaches $2$. So $\lim_{x\to1^{+}}h(x)=2$.

Step7: Analyze limit as $x\to4$

The left - hand limit and right - hand limit as $x\to4$ are different. $\lim_{x\to4^{-}}h(x)=-\infty$ and $\lim_{x\to4^{+}}h(x)=-\infty$. But we consider the general limit concept. Since the function goes to $-\infty$ from both sides, $\lim_{x\to4}h(x)=-\infty$.

Step8: Analyze left - hand limit as $x\to4$

As $x$ approaches $4$ from the left side, the function value goes to $-\infty$. So $\lim_{x\to4^{-}}h(x)=-\infty$.

Step9: Analyze right - hand limit as $x\to4$

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

Step10: Analyze $h(1)$

From the graph, when $x = 1$, the function value $h(1)=1$.

Answer:

(a) $1$
(b) $1$
(c) $1$
(d) DNE
(e) $1$
(f) $2$
(g) $-\infty$
(h) $-\infty$
(i) $-\infty$
(j) $1$