Question

Calculating limits exactly Use the definition of the derivative to evaluate the following limits. $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$$

Short answer

Based on the step-by-step solution, provide a short answer to the problem. The limit, $$\lim_{x \rightarrow e}\frac{\ln x - 1}{x - e}$$, is equal to the derivative of the natural logarithm function evaluated at x=e: $$\frac{1}{e}$$

Step-by-step solution

Recall the definition of the derivative

The definition of the derivative for a function f(x) is given by: $$f'(x) = \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

Determine f(x) and f'(x)

In this problem, we will work with the natural logarithm function, so we have: $$f(x) = \ln x$$ Taking the derivative, we get: $$f'(x) = \lim _{h \rightarrow 0} \frac{\ln(x+h) - \ln(x)}{h}$$

Rewrite the given limit using the definition of the derivative

We are given the limit: $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$$ Let's rewrite this limit in terms of the definition of the derivative: $$f'(e) = \lim _{h \rightarrow 0} \frac{\ln(e+h) - \ln(e)}{h}$$ Notice that \(\ln(e) = 1\). Thus, we have: $$f'(e) = \lim _{h \rightarrow 0} \frac{\ln(e+h) - 1}{h}$$

Substitute and simplify to evaluate the limit

Applying the limit, we get: $$f'(e) = \left.\frac{d}{dx} (\ln x)\right|_{x=e}$$ The derivative of the natural logarithm function is: $$f'(x) = \frac{1}{x}$$ Now, we will substitute x=e: $$f'(e) = \frac{1}{e}$$

Final Answer

So, the limit is equal to the derivative of the natural logarithm function evaluated at x=e: $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e} = \frac{1}{e}$$

Key concepts

Limit Calculation

Understanding the process of limit calculation is crucial when learning calculus. This concept involves figuring out the value that a function approaches as the input (usually represented by the variable 'x') gets arbitrarily close to a specific point. It's like zooming in on a graph at a particular point and observing the value that the function is striving to reach. We represent this mathematically using the limit notation, \( \lim \).

In the provided exercise, we are tasked with evaluating the limit as 'x' approaches the number 'e' for a specific logarithmic function. To solve this, we can relate the given limit to the definition of a derivative, which is itself a limit. By comparing the form of the given limit to the derivative formula and making appropriate substitutions (in this case, replacing the standard 'h' with \( x - e \) and then letting \( h \rightarrow 0 \)), we can proceed to calculate it using the rules of limits and the properties of the natural logarithm.

Natural Logarithm Derivative

Deriving the Natural Logarithm

The natural logarithm derivative is a foundational concept in calculus. The natural logarithm function, denoted as \( \ln(x) \) is special because its base is the constant 'e', which is approximately 2.71828. This irrational number 'e' has many unique properties that make it a natural choice for the logarithm's base in many areas of mathematics, particularly calculus.

The derivative of \( \ln(x) \) is unique in its simplicity: \( \frac{d}{dx}\ln(x) = \frac{1}{x} \). It tells us that for any value of 'x', the slope of the tangent to the curve at that point is the reciprocal of 'x'. When working with derivatives, understanding the relationships between functions and their rates of change is key, and this derivative highlights an essential connection between exponential and logarithmic functions.

Derivative Evaluation

The final step of solving calculus problems often involves derivative evaluation, meaning calculating the derivative value at a specific point. To evaluate the derivative, we apply the derived function formula to a particular 'x' value. This gives us the slope of the tangent line to the curve at that 'x' location.

In our exercise, after deducing the general formula for the derivative of the natural logarithm \( \frac{d}{dx}\ln(x) \), we then focus on a particular point 'e' to find the slope of the function \( \ln(x) \) at this point. Plugging 'e' into our derivative formula \( f'(x) = \frac{1}{x} \) gives us \( f'(e) = \frac{1}{e} \), the final answer we were looking for. Knowing how to evaluate derivatives accurately is vital for understanding the behavior of functions and for solving a wide array of problems in calculus.