Math Problem Statement

Function [h] is graphed. Function h is graphed. The x-axis goes from negative 10 to 10. The graph consists of 2 curves. The first curve starts in quadrant 2, moves downward and ends at an open circle at (negative 1, 3). The second curve starts at an open circle at (negative 1, negative 5), moves upward to about (1, negative 4.8), moves downward to about (7, negative 5.8), moves upward, and ends at about (10, negative 4.2). [\small{1}] [\small{2}] [\small{3}] [\small{4}] [\small{5}] [\small{6}] [\small{7}] [\small{8}] [\small{9}] [\small{\llap{-}2}] [\small{\llap{-}3}] [\small{\llap{-}4}] [\small{\llap{-}5}] [\small{\llap{-}6}] [\small{\llap{-}7}] [\small{\llap{-}8}] [\small{\llap{-}9}] [\small{1}] [\small{2}] [\small{3}] [\small{4}] [\small{5}] [\small{6}] [\small{7}] [\small{8}] [\small{9}] [\small{\llap{-}2}] [\small{\llap{-}3}] [\small{\llap{-}4}] [\small{\llap{-}5}] [\small{\llap{-}6}] [\small{\llap{-}7}] [\small{\llap{-}8}] [\small{\llap{-}9}] [y] [x] [y=h(x)] Select all correct statements about [h] at [x=-1]. Choose all answers that apply: Choose all answers that apply: (Choice A) Both

[\displaystyle\lim_{x\to -1^{+}}h(x)] and

[\displaystyle\lim_{x\to -1^{-}}h(x)] exist A Both

[\displaystyle\lim_{x\to -1^{+}}h(x)] and

[\displaystyle\lim_{x\to -1^{-}}h(x)] exist (Choice B)

[\displaystyle\lim_{x\to -1}h(x)] exists B

[\displaystyle\lim_{x\to -1}h(x)] exists (Choice C)
[h] is defined at [x=-1] C [h] is defined at [x=-1] (Choice D)
[h] is continuous at [x=-1] D [h] is continuous at [x=-1] (Choice E) None of the above E None of the above

Solution

Let's analyze the behavior of the function h(x)h(x) at x=−1x = -1, given the graph description.

  1. The first curve ends at an open circle at (−1,3)(-1, 3), meaning the function does not include this point (it approaches 3 from the left).
  2. The second curve begins at an open circle at (−1,−5)(-1, -5), meaning the function does not include this point either (it approaches -5 from the right).

Now, we check each statement:

(A) Both lim⁡x→−1−h(x)\displaystyle \lim_{x \to -1^-} h(x) and lim⁡x→−1+h(x)\displaystyle \lim_{x \to -1^+} h(x) exist

  • True. From the graph, the function approaches 33 as xx approaches −1-1 from the left (x→−1−x \to -1^-) and approaches −5-5 as xx approaches −1-1 from the right (x→−1+x \to -1^+). Both one-sided limits exist.

(B) lim⁡x→−1h(x)\displaystyle \lim_{x \to -1} h(x) exists

  • False. For the limit to exist at x=−1x = -1, the left-hand limit and right-hand limit must be equal. Since lim⁡x→−1−h(x)=3\displaystyle \lim_{x \to -1^-} h(x) = 3 and lim⁡x→−1+h(x)=−5\displaystyle \lim_{x \to -1^+} h(x) = -5, the two limits are not equal. Thus, the overall limit does not exist.

(C) hh is defined at x=−1x = -1

  • False. The graph shows open circles at both (−1,3)(-1, 3) and (−1,−5)(-1, -5), indicating that h(x)h(x) is not defined at x=−1x = -1.

(D) hh is continuous at x=−1x = -1

  • False. For continuity, h(x)h(x) must be defined at x=−1x = -1, and the limit lim⁡x→−1h(x)\displaystyle \lim_{x \to -1} h(x) must exist and equal h(−1)h(-1). Since h(x)h(x) is not defined at x=−1x = -1 and the limit does not exist, hh is not continuous at x=−1x = -1.

(E) None of the above

  • False. Statement (A) is correct, so this option is not valid.

Correct Answers:

  • (A)

Would you like further clarification or details on limits or continuity? Here are 5 related questions:

  1. What is the difference between one-sided and two-sided limits?
  2. How can a function have one-sided limits but not be continuous at a point?
  3. What are the conditions for a function to be continuous at a point?
  4. How do discontinuities affect the overall behavior of a function?
  5. Can a function be discontinuous at more than one point and still have limits at those points?

Tip: A function's two-sided limit at a point only exists if the left-hand limit and right-hand limit are equal.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Graph Interpretation

Formulas

One-sided limits: lim(x → a⁻) f(x), lim(x → a⁺) f(x)
Two-sided limits: lim(x → a) f(x)

Theorems

Limit Existence Theorem
Continuity Conditions

Suitable Grade Level

Grades 10-12

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