Evaluate The Limit Using L'Hôpital's Rule: Lim ⁡ X → ∞ 3 X 2 E 5 X \lim _{x \rightarrow \infty} \frac{3 X^2}{e^{5 X}} Lim X → ∞ ​ E 5 X 3 X 2 ​

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Introduction

Limits are a fundamental concept in calculus, and they play a crucial role in understanding various mathematical functions. However, evaluating limits can be challenging, especially when dealing with indeterminate forms. In such cases, L'Hôpital's Rule comes to the rescue. In this article, we will explore the application of L'Hôpital's Rule to evaluate the limit of a given function.

What is L'Hôpital's Rule?

L'Hôpital's Rule is a mathematical technique used to evaluate limits of indeterminate forms. It was first introduced by the French mathematician Guillaume de l'Hôpital in the 17th century. The rule states that if a limit is in the form of 0/0 or ∞/∞, we can differentiate the numerator and denominator separately and then take the limit.

Applying L'Hôpital's Rule

To apply L'Hôpital's Rule, we need to follow a specific procedure:

  1. Check if the limit is in the form of 0/0 or ∞/∞: If the limit is in one of these forms, we can proceed with L'Hôpital's Rule.
  2. Differentiate the numerator and denominator separately: We need to find the derivatives of the numerator and denominator with respect to the variable.
  3. Take the limit: After differentiating the numerator and denominator, we take the limit of the resulting expression.

Evaluating the Limit

Now, let's apply L'Hôpital's Rule to evaluate the limit of the given function:

lim⁡x→∞3x2e5x\lim _{x \rightarrow \infty} \frac{3 x^2}{e^{5 x}}

Step 1: Check if the limit is in the form of 0/0 or ∞/∞

As x approaches infinity, the numerator 3x^2 approaches infinity, and the denominator e^(5x) also approaches infinity. Therefore, the limit is in the form of ∞/∞.

Step 2: Differentiate the numerator and denominator separately

To differentiate the numerator, we use the power rule:

ddx(3x2)=6x\frac{d}{dx} (3x^2) = 6x

To differentiate the denominator, we use the chain rule:

ddx(e5x)=5e5x\frac{d}{dx} (e^{5x}) = 5e^{5x}

Step 3: Take the limit

After differentiating the numerator and denominator, we take the limit of the resulting expression:

lim⁡x→∞6x5e5x\lim _{x \rightarrow \infty} \frac{6x}{5e^{5x}}

Simplifying the Expression

To simplify the expression, we can divide both the numerator and denominator by e^(5x):

lim⁡x→∞6x5e5x=lim⁡x→∞6x5e5x⋅1e5x\lim _{x \rightarrow \infty} \frac{6x}{5e^{5x}} = \lim _{x \rightarrow \infty} \frac{6x}{5e^{5x}} \cdot \frac{1}{e^{5x}}

=lim⁡x→∞6x5e10x= \lim _{x \rightarrow \infty} \frac{6x}{5e^{10x}}

Evaluating the Limit

Now, we can evaluate the limit by substituting x = ∞:

lim⁡x→∞6x5e10x=6⋅∞5⋅∞\lim _{x \rightarrow \infty} \frac{6x}{5e^{10x}} = \frac{6 \cdot \infty}{5 \cdot \infty}

Using the property of limits, we can rewrite the expression as:

lim⁡x→∞6x5e10x=65⋅lim⁡x→∞xe10x\lim _{x \rightarrow \infty} \frac{6x}{5e^{10x}} = \frac{6}{5} \cdot \lim _{x \rightarrow \infty} \frac{x}{e^{10x}}

Applying L'Hôpital's Rule Again

The limit is still in the form of ∞/∞, so we can apply L'Hôpital's Rule again:

lim⁡x→∞xe10x=lim⁡x→∞110e10x\lim _{x \rightarrow \infty} \frac{x}{e^{10x}} = \lim _{x \rightarrow \infty} \frac{1}{10e^{10x}}

Evaluating the Limit

Now, we can evaluate the limit by substituting x = ∞:

lim⁡x→∞110e10x=110⋅∞\lim _{x \rightarrow \infty} \frac{1}{10e^{10x}} = \frac{1}{10 \cdot \infty}

Using the property of limits, we can rewrite the expression as:

lim⁡x→∞110e10x=0\lim _{x \rightarrow \infty} \frac{1}{10e^{10x}} = 0

Conclusion

In this article, we applied L'Hôpital's Rule to evaluate the limit of a given function. We started by checking if the limit was in the form of 0/0 or ∞/∞, and then differentiated the numerator and denominator separately. After taking the limit, we simplified the expression and evaluated the limit by substituting x = ∞. The final answer was 0.

Real-World Applications

L'Hôpital's Rule has numerous real-world applications in various fields, including physics, engineering, and economics. For example, it can be used to model population growth, chemical reactions, and economic systems.

Common Mistakes

When applying L'Hôpital's Rule, it's essential to avoid common mistakes, such as:

  • Not checking if the limit is in the form of 0/0 or ∞/∞
  • Not differentiating the numerator and denominator correctly
  • Not taking the limit correctly

Conclusion

Q: What is L'Hôpital's Rule?

A: L'Hôpital's Rule is a mathematical technique used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞.

Q: When can I use L'Hôpital's Rule?

A: You can use L'Hôpital's Rule when the limit is in the form of 0/0 or ∞/∞.

Q: How do I apply L'Hôpital's Rule?

A: To apply L'Hôpital's Rule, you need to follow these steps:

  1. Check if the limit is in the form of 0/0 or ∞/∞.
  2. Differentiate the numerator and denominator separately.
  3. Take the limit of the resulting expression.

Q: What are the common mistakes to avoid when applying L'Hôpital's Rule?

A: Some common mistakes to avoid when applying L'Hôpital's Rule include:

  • Not checking if the limit is in the form of 0/0 or ∞/∞
  • Not differentiating the numerator and denominator correctly
  • Not taking the limit correctly

Q: Can I use L'Hôpital's Rule for limits of the form 0/∞ or ∞/0?

A: No, L'Hôpital's Rule is only applicable for limits of the form 0/0 or ∞/∞.

Q: How do I know if a limit is in the form of 0/0 or ∞/∞?

A: You can determine if a limit is in the form of 0/0 or ∞/∞ by substituting the variable with a value that makes the numerator and denominator equal to zero or infinity.

Q: Can I use L'Hôpital's Rule for limits of trigonometric functions?

A: Yes, L'Hôpital's Rule can be used for limits of trigonometric functions, such as sin(x)/x or cos(x)/x.

Q: How do I apply L'Hôpital's Rule for limits of trigonometric functions?

A: To apply L'Hôpital's Rule for limits of trigonometric functions, you need to differentiate the numerator and denominator separately using the chain rule and the product rule.

Q: Can I use L'Hôpital's Rule for limits of exponential functions?

A: Yes, L'Hôpital's Rule can be used for limits of exponential functions, such as e^x/x or ex/x2.

Q: How do I apply L'Hôpital's Rule for limits of exponential functions?

A: To apply L'Hôpital's Rule for limits of exponential functions, you need to differentiate the numerator and denominator separately using the chain rule and the product rule.

Q: What are some real-world applications of L'Hôpital's Rule?

A: L'Hôpital's Rule has numerous real-world applications in various fields, including physics, engineering, and economics. Some examples include:

  • Modeling population growth
  • Chemical reactions
  • Economic systems

Q: Can I use L'Hôpital's Rule for limits of rational functions?

A: Yes, L'Hôpital's Rule can be used for limits of rational functions, such as x^2/x or x3/x2.

Q: How do I apply L'Hôpital's Rule for limits of rational functions?

A: To apply L'Hôpital's Rule for limits of rational functions, you need to differentiate the numerator and denominator separately using the quotient rule and the product rule.

Conclusion

In conclusion, L'Hôpital's Rule is a powerful mathematical technique used to evaluate limits of indeterminate forms. By following the correct procedure and avoiding common mistakes, we can apply L'Hôpital's Rule to solve a wide range of mathematical problems.