Math Problem Statement
Limite quand x tend vers 0
Solution
To evaluate the limit of as , follow these steps:
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Substitute directly to see if the limit can be evaluated directly: The denominator becomes zero, so the function is undefined at , and we need further analysis.
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Factorize and simplify the expression:
- Numerator: has no immediate factorization relevant here.
- Denominator: .
The function becomes:
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Consider the behavior near :
- The term in the denominator indicates that has a vertical asymptote at .
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Left-hand and right-hand limits: Since appears in the function when expanded, the left-hand limit () and right-hand limit () differ in sign (approaching and , respectively).
Conclusion:
The limit does not exist because the left-hand and right-hand limits are not equal. The function exhibits a vertical asymptote at .
Would you like a step-by-step explanation of vertical asymptotes or limits at infinity?
Related Questions:
- What are the vertical asymptotes of a rational function?
- How do you determine the behavior of a function near singularities?
- What happens to the numerator and denominator as ?
- Can the numerator or denominator be factored further?
- What tools can help analyze one-sided limits?
Tip:
Always check both one-sided limits when the denominator approaches zero to determine the behavior of the function!
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Math Problem Analysis
Mathematical Concepts
Limits
Rational Functions
Asymptotes
Factoring
Formulas
Limit evaluation for rational functions
Factoring expressions
Theorems
Limit laws
Behavior of functions near singularities
Suitable Grade Level
Grades 11-12
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