Evgeny.Makarov said:
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.namerequired said:
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.
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.
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.
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.
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.