Evaluate The Following Limit:${ (c) \lim _{x \rightarrow 0}\left[\frac{1}{x \sqrt{1+x}}-\frac{1}{x}\right] }$

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Introduction

Limits are a fundamental concept in calculus, and evaluating them is a crucial skill for any mathematician. In this article, we will focus on evaluating the limit of a given expression, which involves some algebraic manipulations and trigonometric substitutions. We will break down the problem into smaller steps, making it easier to understand and follow along.

The Given Limit

The given limit is:

(c)lim⁡x→0[1x1+x−1x]{ (c) \lim _{x \rightarrow 0}\left[\frac{1}{x \sqrt{1+x}}-\frac{1}{x}\right] }

This limit involves a square root and a fraction, making it a bit more challenging to evaluate. However, with the right approach, we can simplify the expression and find the limit.

Step 1: Simplify the Expression

To simplify the expression, we can start by combining the two fractions:

1x1+x−1x=1x1+x−xx1+x{ \frac{1}{x \sqrt{1+x}}-\frac{1}{x} = \frac{1}{x \sqrt{1+x}} - \frac{x}{x \sqrt{1+x}} }

Now, we can combine the fractions by finding a common denominator:

1x1+x−xx1+x=1−xx1+x{ \frac{1}{x \sqrt{1+x}} - \frac{x}{x \sqrt{1+x}} = \frac{1 - x}{x \sqrt{1+x}} }

This simplification makes the expression easier to work with.

Step 2: Rationalize the Denominator

To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator:

1−xx1+x⋅1+x1+x=(1−x)1+xx(1+x){ \frac{1 - x}{x \sqrt{1+x}} \cdot \frac{\sqrt{1+x}}{\sqrt{1+x}} = \frac{(1 - x)\sqrt{1+x}}{x(1+x)} }

This step helps to eliminate the square root in the denominator.

Step 3: Simplify the Expression Further

Now, we can simplify the expression further by expanding the numerator:

(1−x)1+xx(1+x)=1+x−x1+xx(1+x){ \frac{(1 - x)\sqrt{1+x}}{x(1+x)} = \frac{\sqrt{1+x} - x\sqrt{1+x}}{x(1+x)} }

This simplification makes the expression easier to work with.

Step 4: Evaluate the Limit

Now, we can evaluate the limit by substituting x = 0 into the simplified expression:

lim⁡x→0[1+x−x1+xx(1+x)]{ \lim _{x \rightarrow 0}\left[\frac{\sqrt{1+x} - x\sqrt{1+x}}{x(1+x)}\right] }

As x approaches 0, the numerator approaches 0, and the denominator approaches 0. However, the numerator approaches 0 faster than the denominator, so the limit is:

lim⁡x→0[1+x−x1+xx(1+x)]=00{ \lim _{x \rightarrow 0}\left[\frac{\sqrt{1+x} - x\sqrt{1+x}}{x(1+x)}\right] = \frac{0}{0} }

This is an indeterminate form, so we need to use L'Hopital's rule to evaluate the limit.

Applying L'Hopital's Rule

L'Hopital's rule states that if a limit is in the form 0/0, we can differentiate the numerator and denominator separately and then evaluate the limit:

lim⁡x→0[1+x−x1+xx(1+x)]=lim⁡x→0[121+x−1+x1+x]{ \lim _{x \rightarrow 0}\left[\frac{\sqrt{1+x} - x\sqrt{1+x}}{x(1+x)}\right] = \lim _{x \rightarrow 0}\left[\frac{\frac{1}{2\sqrt{1+x}} - \sqrt{1+x}}{1+x}\right] }

This simplification makes the expression easier to work with.

Evaluating the Limit

Now, we can evaluate the limit by substituting x = 0 into the simplified expression:

lim⁡x→0[121+x−1+x1+x]{ \lim _{x \rightarrow 0}\left[\frac{\frac{1}{2\sqrt{1+x}} - \sqrt{1+x}}{1+x}\right] }

As x approaches 0, the numerator approaches 0, and the denominator approaches 1. Therefore, the limit is:

lim⁡x→0[121+x−1+x1+x]=12{ \lim _{x \rightarrow 0}\left[\frac{\frac{1}{2\sqrt{1+x}} - \sqrt{1+x}}{1+x}\right] = \frac{1}{2} }

This is the final answer to the given limit.

Conclusion

Introduction

In our previous article, we evaluated the limit of a given expression using algebraic manipulations and trigonometric substitutions. However, we know that limits can be a challenging topic, and many students struggle to understand and apply the concepts. In this article, we will provide a Q&A guide to help you better understand limits and how to evaluate them.

Q: What is a limit?

A: A limit is a value that a function approaches as the input (or independent variable) gets arbitrarily close to a certain point. In other words, it is the value that the function gets arbitrarily close to as the input gets arbitrarily close to a certain point.

Q: Why are limits important?

A: Limits are important because they help us understand how functions behave as the input gets arbitrarily close to a certain point. This is crucial in calculus, as many mathematical concepts, such as derivatives and integrals, rely heavily on limits.

Q: What are some common types of limits?

A: There are several types of limits, including:

  • One-sided limits: These are limits that approach a certain point from one side only.
  • Two-sided limits: These are limits that approach a certain point from both sides.
  • Infinite limits: These are limits that approach infinity as the input gets arbitrarily close to a certain point.
  • Indeterminate forms: These are limits that result in an indeterminate form, such as 0/0 or ∞/∞.

Q: How do I evaluate a limit?

A: Evaluating a limit involves several steps, including:

  • Simplifying the expression: This involves combining like terms and canceling out any common factors.
  • Rationalizing the denominator: This involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate any square roots.
  • Applying L'Hopital's rule: This involves differentiating the numerator and denominator separately and then evaluating the limit.

Q: What is L'Hopital's rule?

A: L'Hopital's rule is a powerful tool for evaluating limits in the form 0/0 or ∞/∞. It states that if a limit is in one of these forms, we can differentiate the numerator and denominator separately and then evaluate the limit.

Q: How do I apply L'Hopital's rule?

A: To apply L'Hopital's rule, follow these steps:

  • Check if the limit is in the form 0/0 or ∞/∞: If it is, then L'Hopital's rule applies.
  • Differentiate the numerator and denominator separately: This will give you a new expression that is easier to evaluate.
  • Evaluate the limit: Once you have differentiated the numerator and denominator, you can evaluate the limit.

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

A: Some common mistakes to avoid when evaluating limits include:

  • Not simplifying the expression: Make sure to simplify the expression before evaluating the limit.
  • Not rationalizing the denominator: Make sure to rationalize the denominator to eliminate any square roots.
  • Not applying L'Hopital's rule when necessary: Make sure to apply L'Hopital's rule when the limit is in the form 0/0 or ∞/∞.

Conclusion

In this article, we provided a Q&A guide to help you better understand limits and how to evaluate them. We covered common types of limits, how to evaluate a limit, and how to apply L'Hopital's rule. We also discussed common mistakes to avoid when evaluating limits. With practice and patience, you can master the art of evaluating limits and become proficient in calculus.