Math Problem Statement
Solution
Let's analyze the content of the image to solve the problem.
The image contains a table with values of and corresponding , followed by three multiple-choice questions asking about the limit of as approaches 3 from different directions:
Table Data:
- ,
- ,
- ,
- ,
- ,
- ,
Questions:
- (Left-hand limit)
- Options: , , , Undefined.
- (Right-hand limit)
- Options: , , , Undefined.
- (Two-sided limit)
- Options: , , , Undefined.
Solving the Limits:
-
Left-hand limit :
- As approaches 3 from the left (, , ), values are getting closer to .
- Answer:
-
Right-hand limit :
- As approaches 3 from the right (, , ), values are getting closer to as well.
- Answer:
-
Two-sided limit :
- Since both the left-hand limit and right-hand limit as approaches 3 are equal to , the two-sided limit exists and is .
- Answer:
Summary of Answers:
- = 6
- = 6
- = 6
Would you like any further details or have any questions?
Related Questions:
- What is the significance of the left-hand and right-hand limits in determining the overall limit?
- How would the answers change if the left-hand and right-hand limits were different?
- What is the relationship between continuity and limits?
- Can a function have different left-hand and right-hand limits? If so, what does it imply?
- How does the concept of limits apply to real-world scenarios?
Tip:
When evaluating limits from a table, always observe how the function's value behaves as it approaches the point of interest from both sides to ensure accuracy.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Functions
Table of Values
Formulas
Limit from the left \(\lim\limits_{x \to c^-} f(x)\)
Limit from the right \(\lim\limits_{x \to c^+} f(x)\)
Two-sided limit \(\lim\limits_{x \to c} f(x)\)
Theorems
Limit definition
Existence of limit
Suitable Grade Level
Grades 10-12