Math Problem Statement
Solution
To solve this problem, we need to determine the value of given that the function is continuous for all real numbers.
Step-by-Step Solution
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Simplify the Expression for :
The given function is:
Notice that the expression is undefined at because the denominator becomes zero. To simplify the expression, factor the numerator:
Thus, we can rewrite as:
For , the terms cancel out:
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Find the Limit as :
Since must be continuous for all real numbers, including at , the limit of as must equal . Calculate the limit:
Substitute :
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Determine the Value of :
For the function to be continuous at , we must have:
Conclusion
The value of is . Thus, the correct answer is:
Would you like further details, or do you have any questions?
Here are 5 related questions for further exploration:
- How would you define continuity of a function at a point?
- What happens if a function is not continuous at a point?
- How do you find the limit of a rational function as it approaches a point of discontinuity?
- Can a function be continuous but not differentiable at some point?
- What is the significance of factoring the numerator and denominator in rational functions?
Tip: Always check for continuity and differentiability separately; continuity does not always imply differentiability!
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Factoring
Rational Functions
Formulas
f(x) = (x^2 - 4) / (x + 2)
Factoring: x^2 - 4 = (x - 2)(x + 2)
Limit: lim(x → -2) (x - 2) = -4
Theorems
Continuity of a function
Limit evaluation for rational functions
Suitable Grade Level
Grades 11-12