Problem 26 - APEX Calculus

Question

Evaluate the given limit. $$ \lim _{x \rightarrow 0} \ln (1+x) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Problem

You are tasked with finding the limit as \( x \) approaches 0 of the function \( \ln(1+x) \). The goal is to evaluate where this function converges as \( x \) gets very close to 0.

Step 2: Recognize the Limit Form

The function \( \ln(1+x) \) is continuous at \( x = 0 \). Recognizing this, we understand that simply substituting \( x = 0 \) into the function gives us a valid result, since the natural logarithm function is defined at this point for \( 1+x \geq 0 \).

Step 3: Substitution

Substitute \( x = 0 \) directly into the function: \( \ln(1+0) = \ln(1) \). Recall that the natural logarithm of 1 is 0.

Step 4: Evaluate the Natural Logarithm

The natural logarithm \( \ln(1) \) evaluates to 0. Therefore, the limit of \( \ln(1+x) \) as \( x \) approaches 0 is 0.

Step 5: Conclusion

Since substituting the limit value (\( x \to 0 \)) into the natural logarithm yields a valid result, the evaluated limit is confirmed.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm with the base \( e \). The number \( e \) is approximately equal to 2.71828 and is considered one of the fundamental constants in mathematics. The natural logarithm provides information about the power to which \( e \) must be raised to obtain the number \( x \). To illustrate, if \( e^y = x \), then \( y = \ln(x) \). This means \( \ln(x) \) is the inverse operation of the exponential function \( e^x \). Here are a few important aspects to remember about natural logarithms:- The domain of \( \ln(x) \) consists of all positive real numbers \( x > 0 \).- \( \ln(1) = 0 \) because \( e^0 = 1 \).- The natural logarithm increases as \( x \) increases. Understanding natural logarithms is essential in evaluating limits involving expressions like \( \ln(1+x) \), especially as \( x \) nears zero. As in our exercise, direct substitution into the function leads to a simple limit evaluation when the argument remains non-negative.

Continuous Functions

A function is considered continuous at a point if its graph does not experience any breaks, jumps, or holes at that point. More formally, a function \( f(x) \) is continuous at a point \( x = a \) if:

• \( f(a) \) is defined, meaning it has a real number value at \( a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The value of the function at \( a \) equals the limit of the function as \( x \) approaches \( a \), that is, \( \lim_{x \to a} f(x) = f(a) \).

In our given problem, \( \ln(1+x) \) is continuous at \( x = 0 \). This is because:

• \( \ln(1+0) = \ln(1) = 0 \), which is defined.
• The limit \( \lim_{x \to 0} \ln(1+x) \) exists and equals \( 0 \).
• The function's value and the limit are equal at this point.

Continuity allows us to confidently substitute \( x = 0 \) directly into the function for accurate limit evaluation.

Limit Evaluation

Evaluating a limit involves finding the value that a function approaches as the variable nears a certain point. It is central to calculus and helps us understand the behavior of functions as they draw near certain points or infinity.When faced with a limit problem like \( \lim_{x \to 0} \ln(1+x) \), follow these steps:

• Understand the form: Determine if substitution directly into the function will provide a defined result.
• Check continuity: Ensure the function is continuous at the point of interest. If it is, simple substitution often suffices.
• Substitute and Simplify: Insert the value into the function and calculate, as with \( \ln(1+0) = \ln(1) = 0 \).
• Conclude: Provide the result and confirm it aligns with your initial steps.

These steps assure understanding and accuracy when evaluating limits. In our case, the calculated limit is \( 0 \). This demonstrates the simplicity of direct approaches when functions are continuous and defined at the point of interest.