Question

Evaluate the limit: \(\lim _{x \rightarrow e} \ln x\).

Short answer

The limit is 1.

Step-by-step solution

Understand the Limit

The problem asks us to evaluate the limit \( \lim_{x \to e} \ln x \). This means we need to determine the value that \( \ln x \) approaches as \( x \) approaches \( e \).

Recall the Properties of natural logarithm

The natural logarithm function \( \ln x \) is continuous for all \( x > 0 \), which means it doesn't have any breaks or jumps in its value. This property is crucial because it allows us to directly substitute \( x = e \) into the function when \( x \) approaches \( e \).

Apply direct substitution

Since \( \ln x \) is continuous at \( x = e \), we can directly substitute \( x = e \) into \( \ln x \) to find the limit. So, \( \lim_{x \to e} \ln x = \ln e \).

Evaluate \(\ln e\)

The natural logarithm of \( e \), \( \ln e \), is a known value. By definition of the natural logarithm, \( \ln e = 1 \). This is because \( e^1 = e \).

State the Conclusion

Therefore, based on the continuity of \( \ln x \) and direct substitution, \( \lim_{x \to e} \ln x = 1 \).

Key concepts

Natural Logarithm

The natural logarithm, symbolized as \( \ln x \), is a logarithm with the base \( e \), where \( e \) is approximately 2.718. This number is a mathematical constant that is key to the concept of continuous growth, appearing in various branches of mathematics and science.
The function \( \ln x \) is defined for all positive values of \( x \) and is a fundamental tool in calculus and analysis. It helps in solving equations involving exponential growth and decay.
Some important properties of the natural logarithm include:

• \( \ln 1 = 0 \): because \( e^0 = 1 \).
• \( \ln e = 1 \): by definition, since \( e^1 = e \).
• The function \( \ln x \) is continuous and differentiable for \( x > 0 \).
• \( \ln(xy) = \ln x + \ln y \)
• \( \ln(x/y) = \ln x - \ln y \)
• \( \ln(x^a) = a \ln x \)

Understanding the natural logarithm and its properties is essential when working with limits and other calculus concepts.

Continuity

In mathematics, continuity describes a function that is smooth and without any breaks, jumps, or holes over an interval.
A function \( f(x) \) is said to be continuous at a point \( a \) if the limit as \( x \) approaches \( a \) is equal to \( f(a) \). In simple terms, you can draw the graph of \( f(x) \) around \( x = a \) without lifting your pencil.
For the natural logarithm function \( \ln x \), it is continuous for all \( x > 0 \). This property allows us to evaluate limits by direct substitution, knowing that the behavior at the point of interest matches the behavior as we approach that point from both sides.
With the limit \( \lim_{x \to e} \ln x \), continuity assures us that \( \ln x \) has no breaks or jumps at \( x = e \), and hence we can safely substitute \( e \) into the function to directly find \( \ln e = 1 \).Continuity is a vital concept that ensures predictability and stability in mathematical functions, making it easier to perform calculus operations.

Direct Substitution

Direct substitution is a method used in calculus to evaluate the limit of a function, where you replace the variable with the value it is approaching.
For functions that are continuous at a certain point, like \( \ln x \) at \( x = e \), the value of the limit can be directly found by simply substituting \( x = e \) into the function.
This process saves time and effort, especially when dealing with well-behaved functions. To ascertain if direct substitution is applicable, it’s crucial first to confirm the continuity of the function at the point in question.
In our example of evaluating \( \lim_{x \to e} \ln x \), by knowing that \( \ln x \) is continuous, we can directly substitute \( e \) into the function and find that \( \ln e = 1 \). This straightforward approach is often the quickest way to solve limits when applicable, keeping calculations simple and efficient.