Question

\(\lim _{\mathrm{x} \rightarrow 1}\left[\left\\{^{\mathrm{n}} \sum_{\mathrm{i}=1}\left(\mathrm{x}^{1}-1\right)\right\\} /(\mathrm{x}-1)\right]=?\) (a) \(n\) (b) \([\\{n(n-1)\\} / 2]\) (c) \([\\{n(n+1)\\} / 2]\) (d) \([(n+1) / 2]\)

Short answer

The short answer is: \(\lim_{x \rightarrow 1}\left[\frac{\sum_{i=1}^{n}(x^{1}-1)}{(x-1)}\right] = n\). The correct answer is (a) n.

Step-by-step solution

Identify the Indeterminate Form

When x approaches 1, the expression becomes indeterminate since both the numerator and the denominator become 0. We will apply L'Hopital's rule to find the limit.

Apply L'Hopital's Rule

According to L'Hopital's rule, if the limit is of the form 0/0 or ∞/∞, then we can differentiate the numerator and the denominator and find their limit. Before applying L'Hopital's rule, let's rewrite the given expression, \(\lim_{x \rightarrow 1} \left[\frac{\sum_{i=1}^{n}(x^{1}-1)}{(x-1)}\right]\) Now, differentiate both the numerator and the denominator with respect to x. The derivative of the numerator is: \(\frac{d}{dx}\left(\sum_{i=1}^{n}(x^{1}-1)\right) = \sum_{i=1}^{n}(1-0) = n\) The derivative of the denominator is: \(\frac{d}{dx}(x-1) = 1\) Now divide the two derivatives and find the limit: \(\lim_{x \rightarrow 1}\frac{n}{1}= n\)

Conclusion

The limit of the expression as x approaches 1 is: \(\lim_{x \rightarrow 1}\left[\frac{\sum_{i=1}^{n}(x^{1}-1)}{(x-1)}\right] = n\) The correct answer is (a) n.

Key concepts

Indeterminate Forms

The concept of indeterminate forms is pivotal in calculus, especially when dealing with limits. An indeterminate form arises when the limit of a mathematical expression cannot be determined directly. These occur in situations such as when both the numerator and the denominator of a fraction approach zero or infinity.

Common indeterminate forms include \(0/0\), \(\infty/\infty\), \(0 \cdot \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\). In the given exercise, when \(x\) approaches 1, we arrive at the \(0/0\) indeterminate form since the numerator and the denominator both become zero. This necessitates the use of advanced techniques such as L'Hopital's Rule to find the limit value.

Differentiation

Differentiation is a fundamental process in calculus, used to determine the rate at which a function is changing at any given point. It's a method of finding the derivative of a function, which essentially represents the slope of the function's graph at a particular point. The act of differentiation transforms a function into its derivative, a new function that provides valuable insights into the original function's behavior.

In the context of solving the limit in the exercise, differentiation is used as part of L'Hopital's Rule. By differentiating the numerator and the denominator separately, we can simplify the expression and avoid the indeterminate form. In our example, differentiating \(\sum_{i=1}^{n}(x-1)\) gives us \(n\), since the derivative of \(x-1\) with respect to \(x\) is 1, and summing this up \(n\) times results in \(n\). Likewise, differentiating the denominator gives us 1. This greatly simplifies the problem, allowing us to directly compute the limit.

Limits in Calculus

Within the realm of calculus, limits are used to describe the behavior of functions as they approach a specific point or infinity. A limit answers the question: 'What value does the function approach as the input approaches a certain value?' The limit does not necessarily equal the function's value at that point, but rather the value it gets arbitrarily close to.

Calculating limits often involves straightforward substitution, but when faced with indeterminate forms such as \(0/0\), we must employ special rules or theorems. L'Hopital's Rule is particularly useful in these scenarios, permitting us to find the limit of a ratio of two functions by taking the limit of the ratio of their derivatives instead. It's especially effective when the original limit yields an indeterminate form. For the exercise presented, L'Hopital’s Rule provides a clear path to solving a limit that, at first glance, seems undetermined.