Problem 379
Question
Evaluate the limits with either L'Hôpital's rule or previously learned methods. \(\lim _{x \rightarrow 1} \frac{x-1}{\ln x}\)
Step-by-Step Solution
Verified Answer
The limit is 1.
1Step 1: Identifying Indeterminate Form
First, let's evaluate the limit directly by substituting the value of 1 into the function. Substituting 1 for x in the numerator results in 0 (since \(1 - 1 = 0\)), and substituting it into the denominator also results in 0 (since \(\ln(1) = 0\)). Thus, the limit is of the form \(\frac{0}{0}\), which allows us to use L'Hôpital's Rule.
2Step 2: Applying L'Hôpital's Rule
According to L'Hôpital's Rule, if a limit is of the indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\). For this problem, \(f(x) = x - 1\) and \(g(x) = \ln x\). Compute the derivatives: - The derivative of \(f(x) = x - 1\) is \(f'(x) = 1\). - The derivative of \(g(x) = \ln x\) is \(g'(x) = \frac{1}{x}\).
3Step 3: Re-evaluating the Limit
Substituting the derivatives into the limit gives: \(\lim_{x \to 1} \frac{1}{\frac{1}{x}} = \lim_{x \to 1} x.\)When we substitute \(x = 1\), the expression simplifies to 1.
Key Concepts
limits in calculusindeterminate formsderivatives in calculus
limits in calculus
In calculus, limits help us understand the behavior of a function as it approaches a particular point. They provide insight into the function's value at points that might not be directly given. When we say \(\lim_{x \to a} f(x) = L\), it means that as \(x\) gets very close to \(a\), \(f(x)\) approaches \(L\). Limits are the foundation for many concepts in calculus, such as derivatives and integrals.
For instance, consider the limit \(\lim_{x \to 1} \frac{x-1}{\ln x}\). At first glance, directly substituting \(x = 1\) gives \(\frac{0}{0}\), which cannot be calculated directly.
This is why techniques like L'Hôpital's Rule are employed. These techniques allow us to navigate around points where direct substitution results in undefined forms, giving us insights into the true nature of the function near those points.
For instance, consider the limit \(\lim_{x \to 1} \frac{x-1}{\ln x}\). At first glance, directly substituting \(x = 1\) gives \(\frac{0}{0}\), which cannot be calculated directly.
This is why techniques like L'Hôpital's Rule are employed. These techniques allow us to navigate around points where direct substitution results in undefined forms, giving us insights into the true nature of the function near those points.
indeterminate forms
An indeterminate form occurs when a limit calculation leads to expressions like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), among others. These forms are termed 'indeterminate' because they don't have a unique value. Instead, they can potentially take multiple values or none at all until more analysis is conducted.
In the problem \(\lim_{x \rightarrow 1} \frac{x-1}{\ln x}\), both the numerator and the denominator become zero as \(x\) approaches 1, leading to the indeterminate form \(\frac{0}{0}\). This is a common scenario when working with logarithmic or polynomial functions.
To resolve indeterminate forms, we can use various methods, such as:
In the problem \(\lim_{x \rightarrow 1} \frac{x-1}{\ln x}\), both the numerator and the denominator become zero as \(x\) approaches 1, leading to the indeterminate form \(\frac{0}{0}\). This is a common scenario when working with logarithmic or polynomial functions.
To resolve indeterminate forms, we can use various methods, such as:
- Algebraic manipulation, to simplify the expression
- L'Hôpital's Rule, which allows us to instead evaluate the limit of the derivatives of the numerator and the denominator
- Considering the function's behavior through graphical or numerical approaches
derivatives in calculus
Derivatives play a crucial role in calculus. They describe how a function changes as its input changes. Specifically, the derivative of a function at a certain point quantifies the rate of change, or the slope of the tangent line at that point. This fundamental concept is used in a variety of applications, from physics to engineering.
L'Hôpital’s Rule leverages derivatives to solve limits that yield indeterminate forms. When we encounter an expression like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), taking the derivative of the numerator and the denominator separately can often clear the indeterminacy.
In the exercise \(\lim_{x \rightarrow 1} \frac{x-1}{\ln x}\), we calculated the derivatives of both components:
Thus, understanding derivatives not only helps in grappling with rate-of-change problems but also provides powerful tools for tackling limits in calculus.
L'Hôpital’s Rule leverages derivatives to solve limits that yield indeterminate forms. When we encounter an expression like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), taking the derivative of the numerator and the denominator separately can often clear the indeterminacy.
In the exercise \(\lim_{x \rightarrow 1} \frac{x-1}{\ln x}\), we calculated the derivatives of both components:
- The derivative of \(f(x) = x - 1\) is \(f'(x) = 1\)
- The derivative of \(g(x) = \ln x\) is \(g'(x) = \frac{1}{x}\)
Thus, understanding derivatives not only helps in grappling with rate-of-change problems but also provides powerful tools for tackling limits in calculus.
Other exercises in this chapter
Problem 378
For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 0} \frac{\tan x}{\sqrt{x}}
View solution Problem 379
For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 1} \frac{x-1}{\ln x} $$
View solution Problem 380
For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 0}(x+1)^{1 / x} $$
View solution Problem 380
Evaluate the limits with either L'Hôpital's rule or previously learned methods. \(\lim _{x \rightarrow 0}(x+1)^{1 / x}\)
View solution