Area under curves and Limit of a sequence,

In summary, the conversation discusses different methods for finding limits, including substitution, expansion, and l'Hospital's rule. The importance of representing the function as a ratio of two functions that tend to the same value is also mentioned.
  • #1
namerequired
4
0
Hello, I am looking for an help about this, I have very short time to do many of them and those are an example, could someone show me one solution or explain me how to do it?
Thank you if you can help me, I really appreciate.
Francesco.

 
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  • #2
Judging by the picture (clickable)

[GRAPH]5wib4rocqz[/GRAPH]

the area is
\[
\int_{0}^2(-(x-2)^3+2)-(x^2-2)\,dx
\]

Limits of the form $\lim_{n\to\infty}f(n)^{g(n)}$ are usually easier to compute when the function is represented as $e^{g(n)\ln(f(n))}$. Which ways of finding limits do you know?
 
  • #3
Evgeny.Makarov said:
Judging by the picture (clickable)

[GRAPH]5wib4rocqz[/GRAPH]

the area is
\[
\int_{0}^2(-(x-2)^3+2)-(x^2-2)\,dx
\]

Limits of the form $\lim_{n\to\infty}f(n)^{g(n)}$ are usually easier to compute when the function is represented as $e^{g(n)\ln(f(n))}$. Which ways of finding limits do you know?

By substitution, or expanding, or Hospital rule i suppose.

Thank you
 
  • #4
namerequired said:
By substitution, or expanding, or Hospital rule i suppose.
I think, the easiest way is to expand $\ln(1+x)$ as $x+o(x)$, but l'Hospital's rule works too. Recall that to apply the rule you need to represent the function as a ratio of two functions that tend both to zero or both to infinity.
 

FAQ: Area under curves and Limit of a sequence,

What is the definition of "area under curves"?

The area under a curve is the measure of the region bounded by the curve and the x-axis. It is typically calculated using integration and can represent quantities such as distance, velocity, or volume.

How is the area under a curve calculated?

The area under a curve is calculated using integration, which involves breaking the region into smaller, known shapes (such as rectangles) and summing their areas. As the number of shapes increases, the accuracy of the calculation also increases.

What is the significance of the limit of a sequence in mathematics?

The limit of a sequence is a fundamental concept in calculus that helps us understand the behavior of a sequence as it approaches a certain value or point. It is used to define continuity, derivatives, and integrals, and is an important tool in determining the convergence or divergence of a series.

How is the limit of a sequence calculated?

The limit of a sequence is calculated by observing the pattern of its terms and finding the value that the terms approach as the sequence continues. This can be done analytically or graphically, using techniques such as the squeeze theorem or the use of limits laws.

What are some real-life applications of the concepts of area under curves and limit of a sequence?

The concepts of area under curves and limit of a sequence are used in various fields such as physics, engineering, economics, and biology. They can help in calculating the distance traveled by an object, determining the efficiency of a process, predicting population growth, and many other real-world scenarios.

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