Questions: Graph f(x) = x^3, x ≠ -2; 0, x = -2 Find lim x → -2^- f(x) and lim x → -2^+ f(x). Does lim x → -2 f(x) exist? If so, what is it? If not, why not? Find lim f(x) and lim f(x). Select the correct choice below and, if necessary, fill in any answer box(es) in your choice. A. lim x → -2^- f(x) = and lim x → -2^+ f(x) does not exist. (Simplify your answer.) B. lim x → -2^- f(x) =, lim x → -2^+ f(x) = (Simplify your answers.) C. lim x → -2^+ f(x) = and lim x → -2^- f(x) does not exist. (Simplify your answer.) D. lim x → -2^- f(x) and lim x → -2^+ f(x) do not exist.

Graph f(x) = x^3, x ≠ -2; 0, x = -2
Find lim x → -2^- f(x) and lim x → -2^+ f(x).
Does lim x → -2 f(x) exist? If so, what is it? If not, why not?
Find lim f(x) and lim f(x). Select the correct choice below and, if necessary, fill in any answer box(es) in your choice.
A. lim x → -2^- f(x) = and lim x → -2^+ f(x) does not exist.
(Simplify your answer.)
B. lim x → -2^- f(x) =, lim x → -2^+ f(x) =
(Simplify your answers.)
C. lim x → -2^+ f(x) = and lim x → -2^- f(x) does not exist.
(Simplify your answer.)
D. lim x → -2^- f(x) and lim x → -2^+ f(x) do not exist.

Transcript text: Graph $f(x)=\left\{\begin{array}{ll}x^{3}, & x \neq-2 \\ 0, & x=-2\end{array}\right.$ Find $\lim _{x \rightarrow-2^{-}} f(x)$ and $\lim _{x \rightarrow-2^{+}} f(x)$. Does $\lim _{x \rightarrow-2} f(x)$ exist? If so, what is it? If not, why not? Find $\lim f(x)$ and $\lim f(x)$. Select the correct choice below and, if necessary, fill in any answer box(es) in your choice. A. $\lim _{x \rightarrow-2^{-}} f(x)=\square$ and $\lim _{x \rightarrow-2^{+}} f(x)$ does not exist. $\square$ (Simplify your answer.) B. $\lim _{x \rightarrow-2^{-}} f(x)=\square, \lim _{x \rightarrow-2^{+}} f(x)=\square$ $\square$ $\square$ (Simplify your answers.) C. $\lim _{x \rightarrow-2^{+}} f(x)=\square$ and $\lim _{x \rightarrow-2^{-}} f(x)$ does not exist. $\square$ (Simplify your answer.) D. $\lim _{x \rightarrow-2^{-}} f(x)$ and $\lim _{x \rightarrow-2^{+}} f(x)$ do not exist.

Solution

Solution Steps

Step 1: Define the function

The function $ f(x) $ is defined as: \[ f(x) = \begin{cases} x^3 & \text{if } x \neq -2 \\ 0 & \text{if } x = -2 \end{cases} \]

Step 2: Find the left-hand limit as $ x $ approaches -2

To find $ \lim_{x \to -2^-} f(x) $: \[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} x^3 = (-2)^3 = -8 \]

Step 3: Find the right-hand limit as $ x $ approaches -2

To find $ \lim_{x \to -2^+} f(x) $: \[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} x^3 = (-2)^3 = -8 \]

Final Answer

a. The function $ f(x) $ is defined as: \[ f(x) = \begin{cases} x^3 & \text{if } x \neq -2 \\ 0 & \text{if } x = -2 \end{cases} \]

b. The left-hand limit as $ x $ approaches -2 is: \[ \lim_{x \to -2^-} f(x) = -8 \]

The right-hand limit as $ x $ approaches -2 is: \[ \lim_{x \to -2^+} f(x) = -8 \]

c. Since both the left-hand limit and the right-hand limit as $ x $ approaches -2 are equal, the limit exists and is: \[ \lim_{x \to -2} f(x) = -8 \]

{"axisType": 3, "coordSystem": {"xmin": -3, "xmax": 3, "ymin": -10, "ymax": 10}, "commands": ["y = x**3"], "latex_expressions": ["$f(x)=\\begin{cases} x^3 & x \\neq -2 \\\\ 0 & x = -2 \\end{cases}$"]}

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