Good Explanation Why $ \lim_{x\to 0^+}\frac{x}{\vert X\vert} $ Is $1$ And Not $0/0$

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Introduction

Limits are a fundamental concept in calculus, and they play a crucial role in understanding the behavior of functions as the input values approach a specific point. In this article, we will delve into the limit of a fraction as x approaches 0 from the right, and we will provide a clear explanation of why the result is 1 and not 0/0.

The Limit in Question

The limit we are interested in is:

lim⁡x→0+x∣x∣\lim_{x\to 0^+}\frac{x}{\vert x\vert}

This limit represents the value that the function approaches as x gets arbitrarily close to 0 from the right. The student who asked the question was expecting the result to be 0/0, but the actual result is 1. In this article, we will explore why this is the case.

Why the Result is Not 0/0

At first glance, it may seem like the limit should be 0/0, since both the numerator and the denominator approach 0 as x approaches 0 from the right. However, this is not the case. To understand why, we need to consider the properties of absolute value.

Absolute Value

For any real number x, the absolute value of x is defined as:

∣x∣={xif x≥0−xif x<0\vert x\vert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

In other words, the absolute value of x is x itself if x is non-negative, and it is -x if x is negative.

Applying Absolute Value to the Limit

Now, let's apply the definition of absolute value to the limit we are interested in:

lim⁡x→0+x∣x∣\lim_{x\to 0^+}\frac{x}{\vert x\vert}

Since x is approaching 0 from the right, we know that x is positive. Therefore, we can use the definition of absolute value to rewrite the limit as:

lim⁡x→0+xx=lim⁡x→0+1\lim_{x\to 0^+}\frac{x}{x} = \lim_{x\to 0^+}1

As x approaches 0 from the right, the value of the function approaches 1, not 0/0.

Conclusion

In conclusion, the limit of a fraction as x approaches 0 from the right is 1, not 0/0. This result may seem counterintuitive at first, but it can be explained by considering the properties of absolute value. By applying the definition of absolute value to the limit, we can see that the result is 1, not 0/0.

Additional Examples

To further illustrate this concept, let's consider a few additional examples.

Example 1: Limit of a Fraction as x Approaches 0 from the Left

Consider the limit:

lim⁡x→0−x∣x∣\lim_{x\to 0^-}\frac{x}{\vert x\vert}

Using the definition of absolute value, we can rewrite this limit as:

lim⁡x→0−x−x=lim⁡x→0−−1\lim_{x\to 0^-}\frac{x}{-x} = \lim_{x\to 0^-}-1

As x approaches 0 from the left, the value of the function approaches -1, not 0/0.

Example 2: Limit of a Fraction as x Approaches 0 from the Right with a Negative Denominator

Consider the limit:

lim⁡x→0+x−x\lim_{x\to 0^+}\frac{x}{-x}

Using the definition of absolute value, we can rewrite this limit as:

lim⁡x→0+xx=lim⁡x→0+1\lim_{x\to 0^+}\frac{x}{x} = \lim_{x\to 0^+}1

As x approaches 0 from the right, the value of the function approaches 1, not 0/0.

Example 3: Limit of a Fraction as x Approaches 0 from the Left with a Negative Denominator

Consider the limit:

lim⁡x→0−x−x\lim_{x\to 0^-}\frac{x}{-x}

Using the definition of absolute value, we can rewrite this limit as:

lim⁡x→0−x−x=lim⁡x→0−−1\lim_{x\to 0^-}\frac{x}{-x} = \lim_{x\to 0^-}-1

As x approaches 0 from the left, the value of the function approaches -1, not 0/0.

Conclusion

Q: What is the limit of a fraction as x approaches 0 from the right?

A: The limit of a fraction as x approaches 0 from the right is 1, not 0/0. This result can be explained by considering the properties of absolute value.

Q: Why is the result not 0/0?

A: The result is not 0/0 because the absolute value of x is x itself if x is non-negative, and it is -x if x is negative. Since x is approaching 0 from the right, we know that x is positive, and therefore, the absolute value of x is x itself.

Q: How does the definition of absolute value apply to the limit?

A: The definition of absolute value applies to the limit by rewriting the fraction as:

lim⁡x→0+x∣x∣=lim⁡x→0+xx=lim⁡x→0+1\lim_{x\to 0^+}\frac{x}{\vert x\vert} = \lim_{x\to 0^+}\frac{x}{x} = \lim_{x\to 0^+}1

Q: What happens if x approaches 0 from the left?

A: If x approaches 0 from the left, the absolute value of x is -x, and therefore, the limit is:

lim⁡x→0−x∣x∣=lim⁡x→0−x−x=lim⁡x→0−−1\lim_{x\to 0^-}\frac{x}{\vert x\vert} = \lim_{x\to 0^-}\frac{x}{-x} = \lim_{x\to 0^-}-1

Q: What if the denominator is negative?

A: If the denominator is negative, the absolute value of x is -x, and therefore, the limit is:

lim⁡x→0+x−x=lim⁡x→0+xx=lim⁡x→0+1\lim_{x\to 0^+}\frac{x}{-x} = \lim_{x\to 0^+}\frac{x}{x} = \lim_{x\to 0^+}1

Q: Can you provide more examples?

A: Yes, here are a few more examples:

Example 1: Limit of a Fraction as x Approaches 0 from the Right with a Negative Denominator

Consider the limit:

lim⁡x→0+x−x\lim_{x\to 0^+}\frac{x}{-x}

Using the definition of absolute value, we can rewrite this limit as:

lim⁡x→0+x−x=lim⁡x→0+xx=lim⁡x→0+1\lim_{x\to 0^+}\frac{x}{-x} = \lim_{x\to 0^+}\frac{x}{x} = \lim_{x\to 0^+}1

Example 2: Limit of a Fraction as x Approaches 0 from the Left with a Negative Denominator

Consider the limit:

lim⁡x→0−x−x\lim_{x\to 0^-}\frac{x}{-x}

Using the definition of absolute value, we can rewrite this limit as:

lim⁡x→0−x−x=lim⁡x→0−x−x=lim⁡x→0−−1\lim_{x\to 0^-}\frac{x}{-x} = \lim_{x\to 0^-}\frac{x}{-x} = \lim_{x\to 0^-}-1

Q: How can I apply this concept to other limits?

A: You can apply this concept to other limits by considering the properties of absolute value and how they apply to the specific limit you are interested in. Remember to always consider the sign of the denominator and how it affects the absolute value of the fraction.

Q: What are some common mistakes to avoid when working with limits?

A: Some common mistakes to avoid when working with limits include:

  • Not considering the sign of the denominator
  • Not applying the definition of absolute value correctly
  • Not simplifying the fraction before taking the limit
  • Not considering the behavior of the function as x approaches the limit point from different directions.

Conclusion

In conclusion, the limit of a fraction as x approaches 0 from the right is 1, not 0/0. This result can be explained by considering the properties of absolute value and how they apply to the specific limit. By understanding these concepts and avoiding common mistakes, you can confidently work with limits and apply them to a wide range of problems.