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:

limx0x31x2\lim_{x \to 0} \frac{x^3 - 1}{x^2}

Let's solve this step by step using algebraic methods and limit theorems.

Step 1: Direct Substitution

First, attempt direct substitution by plugging x=0x = 0 into the expression.

03102=10\frac{0^3 - 1}{0^2} = \frac{-1}{0}

Since division by zero is undefined, we cannot directly substitute x=0x = 0. 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:

x31x2=x3x21x2\frac{x^3 - 1}{x^2} = \frac{x^3}{x^2} - \frac{1}{x^2}

This simplifies further to:

x1x2x - \frac{1}{x^2}

Step 3: Analyze the Behavior as x0x \to 0

Now, evaluate the limit for each term separately.

  • The first term, xx, clearly approaches 0 as x0x \to 0.
  • The second term, 1x2-\frac{1}{x^2}, approaches -\infty as x0x \to 0 because 1x2\frac{1}{x^2} grows without bound when xx is very small and close to 0.

Thus, the entire expression:

x1x2x - \frac{1}{x^2}

will approach -\infty as x0x \to 0.

Step 4: Conclusion

Therefore, the limit does not exist in the usual sense because it tends to -\infty, which indicates that the function diverges as xx approaches 0.

limx0x31x2=\boxed{\lim_{x \to 0} \frac{x^3 - 1}{x^2} = -\infty}

Would you like to go into more details, or have any questions?

Here are 5 related questions for further exploration:

  1. What is the limit of x31x2\frac{x^3 - 1}{x^2} as xx approaches infinity?
  2. How do limits involving infinity differ from finite limits?
  3. What is the behavior of xn1xm\frac{x^n - 1}{x^m} for general nn and mm as xx approaches zero?
  4. How does one handle indeterminate forms like 00\frac{0}{0} using L'Hopital's Rule?
  5. 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