Function $h$ is graphed.

Select all correct statements about $h$ at $x=-1$.

Choose all answers that apply:
A Both $\lim _{x \rightarrow-1^{+}} h(x)$ and $\lim _{x \rightarrow-1^{-}} h(x)$ exist
B $\lim _{x \rightarrow-1} h(x)$ exists
c $h$ is defined at $x=-1$
D $h$ is continuous at $x=-1$
E None of the above

Q: Function $h$ is graphed. Select all correct statements about $h$ at $x=-1$. Choose all answers that apply: A Both $\lim _{x \rightarrow-1^{+}} h(x)$ and $\lim _{x \rightarrow-1^{-}} h(x)$ exist B $\lim _{x \rightarrow-1} h(x)$ exists c $h$ is defined at $x=-1$ D $h$ is continuous at $x=-1$ E None of the above

∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $ \lim_{x \to -1^-} h(x) $ and $ \lim_{x \to -1^+} h(x) $ exist, we observe the graph. The graph approaches the point (-1, -5) from both the left and the right, indicating that both one-sided limits exist. ‖ ‖ step 2 ⋮ To determine if $ \lim_{x \to -1} h(x) $ exists, we need both one-sided limits to be equal. Since both one-sided limits approach -5, $ \lim_{x \to -1} h(x) = -5 $ exists. ‖ ‖ step 3 ⋮ To determine if $ h $ is defined at $ x = -1 $, we observe the graph. There is a hole at the point (-1, -5), indicating that $ h(-1) $ is not defined. ‖ ‖ step 4 ⋮ To determine if $ h $ is continuous at $ x = -1 $, the function must be defined at $ x = -1 $ and the limit as $ x $ approaches -1 must equal $ h(-1) $. Since $ h(-1) $ is not defined, $ h $ is not continuous at $ x = -1 $. ‖ ***A*** ∻Key Concept∻ ⚹Existence of Limits⚹ ∻Explanation∻ ⚹For a limit to exist at a point, both one-sided limits must exist and be equal.⚹◊What is the definition of continuity of a function at a point in calculus⍭ Generate me a similar question◊

Question

Function $h$ is graphed.

Select all correct statements about $h$ at $x=-1$.

Choose all answers that apply:
A Both $\lim _{x \rightarrow-1^{+}} h(x)$ and $\lim _{x \rightarrow-1^{-}} h(x)$ exist
B $\lim _{x \rightarrow-1} h(x)$ exists
c $h$ is defined at $x=-1$
D $h$ is continuous at $x=-1$
E None of the above

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $ \lim_{x o -1^-} h(x) $ and $ \lim_{x o -1^+} h(x) $ exist, we observe the graph. The graph approaches the point (-1, -5) from both the left and the right, indicating that both one-sided limits exist. ‖ ‖ step 2 ⋮ To determine if $ \lim_{x o -1} h(x) $ exists, we need both one-sided limits to be equal. Since both one-sided limits approach -5, $ \lim_{x o -1} h(x) = -5 $ exists. ‖ ‖ step 3 ⋮ To determine if $ h $ is defined at $ x = -1 $, we observe the graph. There is a hole at the point (-1, -5), indicating that $ h(-1) $ is not defined. ‖ ‖ step 4 ⋮ To determine if $ h $ is continuous at $ x = -1 $, the function must be defined at $ x = -1 $ and the limit as $ x $ approaches -1 must equal $ h(-1) $. Since $ h(-1) $ is not defined, $ h $ is not continuous at $ x = -1 $. ‖ ***A*** ∻Key Concept∻ ⚹Existence of Limits⚹ ∻Explanation∻ ⚹For a limit to exist at a point, both one-sided limits must exist and be equal.⚹◊What is the definition of continuity of a function at a point in calculus⍭ Generate me a similar question◊

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