Math Problem Statement

generate questions on Getting Limiting Values ​ from Graphs​ and show a explained answer key at the end

Solution

Here are five questions related to finding limiting values from graphs, followed by an explained answer key:

Questions:

  1. Question 1: Consider the graph of a function f(x)f(x). As x2x \to 2, the graph approaches a horizontal line at y=3y = 3, but the function is undefined at x=2x = 2. What is the limiting value of f(x)f(x) as xx approaches 2?

  2. Question 2: For the graph of g(x)g(x), it is observed that as xx \to -\infty, the graph gets closer and closer to the horizontal line y=5y = -5. What is the limiting value of g(x)g(x) as xx \to -\infty?

  3. Question 3: The graph of a function h(x)h(x) shows that as x4x \to 4, the graph approaches y=7y = 7 from both sides, but at x=4x = 4, the value of the function is h(4)=2h(4) = 2. What is the limit of h(x)h(x) as x4x \to 4?

  4. Question 4: Looking at the graph of a function k(x)k(x), the function increases without bound as x3+x \to 3^+ (from the right). What is the limit of k(x)k(x) as x3+x \to 3^+?

  5. Question 5: From the graph of p(x)p(x), we see that as x1x \to -1 from the left, the function approaches 0, but as x1x \to -1 from the right, the function approaches 2. What can you say about the limit of p(x)p(x) as x1x \to -1?


Explained Answer Key:

  1. Answer 1: The limit of f(x)f(x) as x2x \to 2 is the value that the function approaches as xx gets closer and closer to 2, regardless of whether the function is defined at x=2x = 2 or not. From the graph, we observe that as x2x \to 2, the function approaches y=3y = 3. Therefore, limx2f(x)=3.\lim_{x \to 2} f(x) = 3.

  2. Answer 2: The limit as xx \to -\infty refers to the behavior of the function as xx decreases without bound. The graph indicates that the function approaches the horizontal line y=5y = -5. Therefore, limxg(x)=5.\lim_{x \to -\infty} g(x) = -5.

  3. Answer 3: The limit of h(x)h(x) as x4x \to 4 depends on the values the function approaches as xx gets closer to 4 from both sides. Although the function is defined at h(4)=2h(4) = 2, this value does not affect the limiting behavior. The graph shows that as xx approaches 4, the function approaches y=7y = 7. Thus, limx4h(x)=7.\lim_{x \to 4} h(x) = 7.

  4. Answer 4: The behavior of the function as x3+x \to 3^+ (from the right) shows that the function increases without bound, which means that the function tends towards positive infinity. Therefore, limx3+k(x)=+.\lim_{x \to 3^+} k(x) = +\infty.

  5. Answer 5: When the left-hand and right-hand limits of a function as x1x \to -1 are not equal, the overall limit at that point does not exist. In this case, the left-hand limit is 00 and the right-hand limit is 22. Since these limits are not the same, we conclude that limx1p(x) does not exist.\lim_{x \to -1} p(x) \text{ does not exist}.


Follow-up Questions:

  1. How do you determine the limit of a function if the function is undefined at the point of interest?
  2. What is the difference between the left-hand limit and right-hand limit of a function?
  3. Can a function have a limit at a point where it is not continuous? Explain why or why not.
  4. How does the behavior of a function as xx \to \infty or xx \to -\infty relate to its asymptotes?
  5. What role do discontinuities play when finding limits from graphs?

Tip: When evaluating limits from graphs, focus on the behavior of the function as it approaches the point from either side, rather than the actual value of the function at that point.

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Math Problem Analysis

Mathematical Concepts

Limits
Graphical Interpretation of Limits
Asymptotic Behavior
Discontinuities

Formulas

lim_{x \to c} f(x)
Left-hand limit: lim_{x \to c^-} f(x)
Right-hand limit: lim_{x \to c^+} f(x)

Theorems

Limit Laws
Continuity and Discontinuity Theorems
Squeeze Theorem

Suitable Grade Level

Grades 11-12