Question

Evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 0} \frac{1+1 / x}{1-1 / x} $$

Short answer

The limit is -1.

Step-by-step solution

Identify the Indeterminate Form

Evaluate the expression as it approaches the limit \(x \to 0\): \(\frac{1+1/x}{1-1/x}\). As \(x\) approaches 0, both the numerator \((1 + 1/x)\) and the denominator \((1 - 1/x)\) become undefined, leading to an indeterminate form \(\frac{\infty}{\infty}\). This indicates L'Hôpital's Rule can be used.

Apply L'Hôpital's Rule

According to L'Hôpital's Rule, if \(\lim_{x \to c} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then: \[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \] Apply this to \(f(x) = 1 + \frac{1}{x}\) and \(g(x) = 1 - \frac{1}{x}\). Calculate the derivatives: \(f'(x) = -\frac{1}{x^2}\) and \(g'(x) = \frac{1}{x^2}\).

Evaluate the Limit After Differentiation

Substitute the derivatives into the limit: \(\lim_{x \to 0} \frac{-1/x^2}{1/x^2}\). Simplify the expression to \(\lim_{x \to 0} -1 = -1\).

Conclude the Limit Evaluation

After applying L'Hôpital's Rule and evaluating the simplified limit, we find that the expression tends towards \(-1\) as \(x \to 0\). This is the final result for the given limit.

Key concepts

Indeterminate Forms

When evaluating limits in calculus, we may encounter expressions that do not initially yield clear results. These expressions are called "indeterminate forms." They occur when substitution leads to forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), which do not provide a definitive limit.
Indeterminate forms indicate that direct substitution is insufficient and a different approach, such as L'Hôpital's Rule, is necessary to find the exact limit. Some common indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( 0 \times \infty \)
• \( \infty - \infty \)
• \( 0^0 \)
• \( \infty^0 \)
• \( 1^\infty \)

Each of these situations requires different techniques to resolve the actual limit value, with L'Hôpital's Rule being particularly useful for fraction forms like \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). Understanding these forms provides the foundation for applying appropriate calculus techniques.

Differentiation

Differentiation is a fundamental concept in calculus used to determine the rate at which a function changes. It involves calculating the derivative of a function, which represents the slope of the tangent line at any given point.
In the context of limits, differentiation is particularly useful when applying L'Hôpital's Rule. L'Hôpital's Rule states that if the limit of a fraction results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), the limit can be found by differentiating the numerator and the denominator separately and then taking the limit again.Here’s how differentiation is applied in L'Hôpital's Rule:

• Identify \( f(x) \) and \( g(x) \) from the limit expression \( \frac{f(x)}{g(x)} \).
• Calculate the derivatives \( f'(x) \) and \( g'(x) \).
• Evaluate \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \) if the form initially was indeterminate.

Differentiation not only helps in solving limits but also provides insights into the behavior and characteristics of functions.

Limit Evaluation

Limit evaluation is a core concept in calculus used to describe the behavior of a function as its argument approaches a particular value. This process is vital for understanding continuous change, predicting function behavior, and solving problems in calculus.The solution involves substituting the value the variable approaches directly into the function. However, when direct substitution results in an indeterminate form, alternative methods like L'Hôpital's Rule are applied.Consider these steps for evaluating limits:

• Check if direct substitution results in a determinate value.
• If substitution gives an indeterminate form, consider methods like L'Hôpital's Rule.
• Apply the rule by differentiating the numerator and denominator, then substitute again.
• Simplify where necessary to find the limit.

In the presented exercise, direct substitution in \( \lim _{x \to 0} \frac{1+1/x}{1-1/x} \) results in \( \frac{\infty}{\infty} \), an indeterminate form. Therefore, we use L'Hôpital's Rule, differentiate the terms, and re-evaluate the limit, leading us to a clear solution of \(-1\).
Through such processes, limit evaluation helps you not only find precise values but also understand the overall behavior of mathematical expressions.