Math Problem Statement
limit of( x cube -1)/x square when x approaches zero.solve using limit theorem and methods by algebraically for existence.
Solution
We are tasked with evaluating the limit:
Let's solve this step by step using algebraic methods and limit theorems.
Step 1: Direct Substitution
First, attempt direct substitution by plugging into the expression.
Since division by zero is undefined, we cannot directly substitute . This suggests that we need a different approach, such as simplifying the expression.
Step 2: Rewrite the Expression
We can simplify the expression by separating the terms of the fraction:
This simplifies further to:
Step 3: Analyze the Behavior as
Now, evaluate the limit for each term separately.
- The first term, , clearly approaches 0 as .
- The second term, , approaches as because grows without bound when is very small and close to 0.
Thus, the entire expression:
will approach as .
Step 4: Conclusion
Therefore, the limit does not exist in the usual sense because it tends to , which indicates that the function diverges as approaches 0.
Would you like to go into more details, or have any questions?
Here are 5 related questions for further exploration:
- What is the limit of as approaches infinity?
- How do limits involving infinity differ from finite limits?
- What is the behavior of for general and as approaches zero?
- How does one handle indeterminate forms like using L'Hopital's Rule?
- How can we graphically interpret the behavior of this function near zero?
Tip: When encountering limits that tend toward infinity or indeterminate forms, simplifying the function or using tools like L'Hopital's Rule can often clarify the behavior of the function.
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Math Problem Analysis
Mathematical Concepts
Limits
Algebraic Simplification
Asymptotic Behavior
Formulas
Limit of (f(x) / g(x)) as x approaches a
Simplification of rational expressions
Theorems
Limit Theorems
L'Hopital's Rule (applicable if desired)
Suitable Grade Level
Grades 11-12 and Undergraduate
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