Question

Find the following limits (a) \(\lim _{x \rightarrow 0} \frac{e-(1+x)^{1 / x}}{x}\). (b) \(\lim _{x \rightarrow \infty} \frac{n}{\log n}\left[n^{1 / n}-1\right]\). (c) \(\lim _{x \rightarrow 0} \frac{\tan x-x}{x(1-\cos x)}\) (d) \(\lim _{x \rightarrow 0} \frac{x-\sin x}{\tan x-x}\)

Short answer

Answer: The limit is \(\ln10\).

Step-by-step solution

Rewrite the Expression

Rewrite the expression as \(\lim _{x \rightarrow 0} \frac{e^1-(1+x)^{1 / x}}{x}\).

Apply L'Hopital's Rule

Differentiate the numerator and denominator and apply L'Hopital's rule: \(\lim _{x \rightarrow 0} \frac{d}{dx}(e^1 - (1+x)^{1/x}) / \frac{d}{dx}(x)\).

Find the Derivatives

The derivative of the numerator is \(0 - \frac{1}{x^2} (1+x)^{1/x-1}\), and the derivative of the denominator is 1.

Evaluate the Limit

Evaluate the limit and simplify: \(\lim _{x \rightarrow 0} -\frac{1}{x^2} (1+x)^{1/x-1} = -\frac{1}{1}e^0 = -1\). The limit is \(-1\). (b) Find \(\lim _{x \rightarrow \infty} \frac{n}{\log n}\left[n^{1 / n}-1\right]\).

Apply L'Hopital's Rule

Differentiate the numerator and denominator and apply L'Hopital's rule: \(\lim _{n \rightarrow \infty} \frac{d}{dn}(n(n^{1/n}-1))/\frac{d}{dn}(\log n)\).

Find the Derivatives

The derivative of the numerator is \((1-\frac{\log n}{n+1})\) and the derivative of the denominator is \(\frac{1}{n\ln10}\).

Evaluate the Limit

Evaluate the limit and simplify: \(\lim _{n \rightarrow \infty} (1-\frac{\log n}{n+1})\frac{n\ln10}{1} = 1\cdot\ln10\). The limit is \(\ln10\). (c) Find \(\lim _{x \rightarrow 0} \frac{\tan x-x}{x(1-\cos x)}\).

Use Trigonometric Limit Identities

Notice that \(\lim_{x \rightarrow 0} \frac{\tan x - x}{x^2} = \frac{1}{3}\) and \(\lim_{x \rightarrow 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\).

Express the Problem As the Multiplication of Two Limits

The given problem is equivalent to \(\lim_{x \rightarrow 0} \frac{\frac{\tan x - x}{x^2}}{\frac{1 - \cos x}{x^2}}\).

Evaluate the Limits

Evaluate the two limits individually: \(\frac{\lim_{x \rightarrow 0} \frac{\tan x - x}{x^2}}{\lim_{x \rightarrow 0} \frac{1 - \cos x}{x^2}}\). This simplifies to \(\frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}\). The limit is \(\frac{2}{3}\). (d) Find \(\lim _{x \rightarrow 0} \frac{x-\sin x}{\tan x-x}\).

Apply L'Hopital's Rule

Differentiate the numerator and denominator and apply L'Hopital's rule: \(\lim _{x \rightarrow 0} \frac{d}{dx}(x-\sin x)/\frac{d}{dx}(\tan x-x)\).

Find the Derivatives

The derivative of the numerator is \(1-\cos x\) and the derivative of the denominator is \(\sec^2 x - 1\).

Evaluate the Limit

Evaluate the limit and simplify: \(\lim _{x \rightarrow 0} \frac{1-\cos x}{\sec^2 x - 1} = \frac{1-1}{1-1}\). The limit is not defined as it results in \(\frac{0}{0}\).

Key concepts

L'Hopital's Rule

When students encounter a limit problem that results in an indeterminate form like \(0/0\) or \(\infty/\infty\), L'Hopital's Rule is a powerful tool for finding the limit. This rule states that if the limits of the functions in the numerator and denominator both approach zero or infinity, we can take the derivatives of these functions and re-evaluate the limit. For instance, in the exercise solution (a) and (d), we applied L'Hopital's Rule because the direct substitution of \(x\) led to the indeterminate form \(0/0\).
Applying L'Hopital's rule correctly, however, requires certain conditions to be met. Firstly, both the functions in the numerator and denominator should be differentiable near the point of interest. And secondly, after taking the derivatives, the limit must resolve to a determinate form or at least lead to a form where further application of the rule is helpful.
It's also crucial to notice that L'Hopital's Rule may need to be applied more than once, and sometimes it may not be the most effective technique. As observed in exercise solution (d), applying this rule blindly can lead to incorrect conclusions. Adequate mathematical analysis is needed to determine the correct approach for solving a limit problem.

Trigonometric Limit Identities

When solving limit problems involving trigonometric functions, recognizing and applying trigonometric limit identities can greatly simplify the process. These identities capture the behavior of trigonometric functions as the argument approaches a specific value. For example, it's known that \(\lim_{x \rightarrow 0} \sin x / x = 1\) and \(\lim_{x \rightarrow 0} (1 - \cos x) / x^2 = 1/2\).
Part (c) of the exercise illustrates how knowing these identities can be essential for solving a limit involving trigonometric functions. By dividing the numerator and the denominator by \(x^2\), the problem is reduced to a known trigonometric limit, which greatly eases the computation. This substitution is a typical strategy to apply when you encounter trigonometric functions within limit problems. These identities not only streamline calculations but also help in understanding the nature and behavior of trigonometric functions around crucial points such as zero.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits, functions, and calculus. It provides a rigorous foundation for understanding changes and motion, which are central concepts in physics and other sciences. In the realm of limits, the goal of mathematical analysis is to evaluate the behavior of functions as they approach specific points or infinity.
Throughout the process of solving limit problems, it's important to consistently conduct a thorough mathematical analysis. This starts with identifying the nature of the problem, such as whether it involves indeterminate forms. After recognizing indeterminate forms, such as in the given exercise solutions, a deeper analysis is needed to decide the suitable method to resolve them, be it L'Hopital's Rule, trigonometric identities, factoring, or any other technique.
Mathematical analysis is not only about finding answers but also validating them. We should always check if the result makes sense in the context of the function's behavior. For example, in exercise solution (d), a further analysis indicates that simply applying L'Hopital’s Rule leads to an undefined limit, which signals the need for a different approach or an alternative method of proof. This meticulous approach ensures the correctness and enhances the understanding of limit problems.