Math Problem Statement
lim x->-3- ([x]/(sign(x+3)))
Solution
To evaluate the limit:
we need to understand the behavior of the floor function and the sign function as approaches from the left.
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Analyzing :
The function gives the sign of . Specifically:- If , is positive, so .
- If , , but we don't evaluate the sign at this exact point due to the limit approaching from the left.
- If , is negative, so .
Since we're approaching from the left (), in this context, and thus .
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Analyzing : The floor function takes the greatest integer less than or equal to .
- As , values of just to the left of (like , , etc.) would have .
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Substitute and Simplify: Given these behaviors, we have:
Therefore, the limit is:
Would you like further clarification on any part of this? Here are some related questions you might explore:
- What is the limit if approaches from the right?
- How does the behavior of the floor function affect limits near integers?
- What are other examples where the sign function is used in limits?
- How would this limit change if we used the ceiling function instead of the floor function?
- Why is the left-hand limit different from a right-hand limit in piecewise functions?
Tip: In limits involving floor functions, consider the integer behavior of around the point of interest—approaching from left or right can yield different integer values.
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Math Problem Analysis
Mathematical Concepts
Limits
Floor Function
Sign Function
Left-Hand Limit
Formulas
\lim_{x \to a} \frac{f(x)}{g(x)}
floor function \(\lfloor x \rfloor\)
sign function \(\operatorname{sign}(x)\)
Theorems
Limit of a function
Piecewise behavior of floor and sign functions
Suitable Grade Level
Grades 11-12
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