Evaluate The Limit Using L'Hôpital's Rule If Necessary: Lim ⁡ X → ∞ ( 9 X 9 X + 2 ) 8 X \lim _{x \rightarrow \infty}\left(\frac{9x}{9x+2}\right)^{8x} Lim X → ∞ ​ ( 9 X + 2 9 X ​ ) 8 X

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Introduction

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In this article, we will evaluate the limit of the expression $\left(\frac{9x}{9x+2}\right)^{8x}$ as $x$ approaches infinity. We will use L'Hôpital's rule if necessary to find the limit.

The Limit Expression

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The given limit expression is $\lim _{x \rightarrow \infty}\left(\frac{9x}{9x+2}\right)^{8x}$. This expression involves a power function and a rational function. To evaluate this limit, we need to simplify the expression and then apply the appropriate limit rules.

Simplifying the Expression

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We can simplify the expression by rewriting it as $\left(\frac{9x}{9x+2}\right)^{8x} = \left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$.

Applying L'Hôpital's Rule

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To evaluate the limit of the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$, we can apply L'Hôpital's rule. L'Hôpital's rule states that if the limit of a function is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then we can differentiate the numerator and denominator separately and take the limit of the resulting expression.

Differentiating the Numerator and Denominator

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Let's differentiate the numerator and denominator of the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$.

The numerator is $1$, so its derivative is $0$.

The denominator is $1+\frac{2}{9x}$, so its derivative is $-\frac{2}{9x^2}$.

Applying L'Hôpital's Rule

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Now that we have differentiated the numerator and denominator, we can apply L'Hôpital's rule. The limit of the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$ is equal to the limit of the expression $\frac{0}{-\frac{2}{9x^2}}$.

Evaluating the Limit

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The limit of the expression $\frac{0}{-\frac{2}{9x^2}}$ is equal to $0$.

Conclusion

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In this article, we evaluated the limit of the expression $\left(\frac{9x}{9x+2}\right)^{8x}$ as $x$ approaches infinity. We simplified the expression and then applied L'Hôpital's rule to find the limit. The final answer is $e^{-\frac{2}{9}}$.

The Final Answer

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The final answer is $\boxed{e^{-\frac{2}{9}}}$.

Step-by-Step Solution

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Here are the step-by-step solutions to the problem:

1. Simplify the expression $\left(\frac{9x}{9x+2}\right)^{8x}$.
2. Apply L'Hôpital's rule to the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$.
3. Differentiate the numerator and denominator of the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$.
4. Apply L'Hôpital's rule to the expression $\frac{0}{-\frac{2}{9x^2}}$.
5. Evaluate the limit of the expression $\frac{0}{-\frac{2}{9x^2}}$.

Frequently Asked Questions

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Here are some frequently asked questions about the problem:

• What is the limit of the expression $\left(\frac{9x}{9x+2}\right)^{8x}$ as $x$ approaches infinity?
• How do we simplify the expression $\left(\frac{9x}{9x+2}\right)^{8x}$?
• What is L'Hôpital's rule, and how do we apply it to the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$?
• How do we differentiate the numerator and denominator of the expression $\left(\frac{1}{1+\frac{2}{9x}}\right)^{8x}$?
• What is the final answer to the problem?

Conclusion

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In this article, we evaluated the limit of the expression $\left(\frac{9x}{9x+2}\right)^{8x}$ as $x$ approaches infinity. We simplified the expression and then applied L'Hôpital's rule to find the limit. The final answer is $e^{-\frac{2}{9}}$.

References

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Here are some references that we used to write this article:

• L'Hôpital's rule
• Differentiation
• Limits
• Exponents
• Rational functions

Further Reading

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Here are some further reading resources that you can use to learn more about the topic:

• Calculus
• Mathematics
• Limits
• Exponents
• Rational functions

Final Answer

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The final answer is $\boxed{e^{-\frac{2}{9}}}$.

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Introduction

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In our previous article, we evaluated the limit of the expression $\left(\frac{9x}{9x+2}\right)^{8x}$ as $x$ approaches infinity using L'Hôpital's rule. In this article, we will answer some frequently asked questions about evaluating limits using L'Hôpital's rule.

Q&A

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Q: What is L'Hôpital's rule, and how do we apply it to evaluate limits?

A: L'Hôpital's rule is a mathematical technique used to evaluate limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. To apply L'Hôpital's rule, we differentiate the numerator and denominator of the expression separately and then take the limit of the resulting expression.

Q: When can we use L'Hôpital's rule to evaluate limits?

A: We can use L'Hôpital's rule to evaluate limits when the limit is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. This means that the numerator and denominator must both approach zero or infinity as the variable approaches a certain value.

Q: How do we differentiate the numerator and denominator of an expression?

A: To differentiate the numerator and denominator of an expression, we apply the power rule and the sum rule of differentiation. For example, if we have the expression $x^2 + 3x$, we can differentiate it using the power rule and the sum rule.

Q: What are some common mistakes to avoid when using L'Hôpital's rule?

A: Some common mistakes to avoid when using L'Hôpital's rule include:

• Not checking if the limit is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying L'Hôpital's rule.
• Not differentiating the numerator and denominator correctly.
• Not taking the limit of the resulting expression after differentiating the numerator and denominator.

Q: Can we use L'Hôpital's rule to evaluate limits of trigonometric functions?

A: Yes, we can use L'Hôpital's rule to evaluate limits of trigonometric functions. For example, we can use L'Hôpital's rule to evaluate the limit of $\sin x$ as $x$ approaches infinity.

Q: Can we use L'Hôpital's rule to evaluate limits of exponential functions?

A: Yes, we can use L'Hôpital's rule to evaluate limits of exponential functions. For example, we can use L'Hôpital's rule to evaluate the limit of $e^x$ as $x$ approaches infinity.

Conclusion

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In this article, we answered some frequently asked questions about evaluating limits using L'Hôpital's rule. We discussed when to use L'Hôpital's rule, how to differentiate the numerator and denominator of an expression, and some common mistakes to avoid when using L'Hôpital's rule.

Further Reading

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Here are some further reading resources that you can use to learn more about the topic:

• Calculus
• Mathematics
• Limits
• Exponents
• Rational functions

Final Answer

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The final answer is $\boxed{e^{-\frac{2}{9}}}$.