Conquering the Infinity Limit: Integrals and Exponential Functions

In summary, the conversation discusses the use of L'Hopital's Rule and other mathematical techniques to simplify the expression lim_{x\rightarrow + ∞} \frac{\int^{x^3}_{0} e^{t^2}dt}{x \int^{x^2}_{0} e^{t^2}dt} by applying the Product Rule, Fundamental Theorem of Calculus, and Chain Rule. The final goal is to find and simplify the inverse of the expression, which will then lead to determining the value of the original expression at the limit.
  • #1
powerof
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[itex]lim_{x\rightarrow + ∞} \frac{\int^{x^3}_{0} e^{t^2}dt}{x \int^{x^2}_{0} e^{t^2}dt} [/itex]

Attempt at a solution: I don't really know where to start. Any hints?
 
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  • #2
L' Hopital's Rule. Product Rule for the denominator. Fundamental Theorem of Calculus to differentiate the integrals. Don't forget that the bounds are functions of ##x##, so apply Chain Rule (Leibniz's Rule).

After the first step, you'll end up with an expression that I'll call ##L##. Find and simplify ##L^{-1}## with a further application of all those rules. Then deduce what ##L## should be at the limit.
 
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  • #3
Thanks. When I get home I'll get to work.
 

1. What is a limit with integrals problem?

A limit with integrals problem is a type of mathematical problem that involves finding the limit of a function as it approaches a certain point, using integrals instead of traditional algebraic methods. This is often seen in calculus courses and can involve both one-dimensional and multi-dimensional functions.

2. How do you solve a limit with integrals problem?

To solve a limit with integrals problem, you first need to set up the integral for the given function. Then, you can use various techniques such as substitution, integration by parts, or trigonometric identities to evaluate the integral. Finally, you can take the limit of the resulting expression to find the solution.

3. What is the purpose of using integrals in limit problems?

Integrals allow us to find the area under a curve, which can be used to determine the limit of a function as it approaches a certain point. This is especially useful for functions that cannot be easily evaluated using traditional algebraic methods.

4. What are some common challenges in solving limit with integrals problems?

Some common challenges in solving limit with integrals problems include identifying the appropriate technique to use, correctly setting up the integral, and evaluating the resulting expression. It is also important to pay attention to the limits of integration and make sure they are consistent with the given function.

5. How can I practice and improve my skills in solving limit with integrals problems?

The best way to practice and improve your skills in solving limit with integrals problems is to work through various examples and exercises. You can also seek help from a tutor or attend study groups to get additional guidance and support. Additionally, understanding the underlying concepts and theory behind integrals can also help improve your problem-solving abilities.

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