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

Answer: The limit of the function as x approaches e is \(\frac{1}{e}\).

Step-by-step solution

Identify if the function is continuous and differentiable

In this case, the function is the natural logarithm of x. Since the natural logarithm function is continuous and differentiable for positive values of x, we can proceed to the next step.

Use the definition of the derivative to rewrite the limit

The definition of the derivative is: $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$ In our case, \(f(x) = \ln x\), \(a = e\), and \(f(a) = \ln e \). Therefore, the limit can be rewritten as: $$f'(e) = \lim_{x \to e} \frac{\ln x - \ln e}{x - e}$$

Simplify the limit expression

Recall that the natural logarithm of e is equal to 1. So, we can rewrite the limit expression as: $$f'(e) = \lim_{x \to e} \frac{\ln x - 1}{x - e}$$

Using L'Hopital's Rule

The limit expression is in the indeterminate form \(\frac{0}{0}\), thus we can apply L'Hopital's Rule. L'Hopital's Rule states that, if the limit approaches an indeterminate form, we can find the limit by taking the derivative of both the numerator and denominator separately and evaluating the limit of the new function. Therefore, $$\lim_{x \rightarrow e} \frac{\ln x-1}{x-e} = \lim_{x \rightarrow e} \frac{(d/dx)(\ln x - 1)}{(d/dx)(x-e)}$$

Find the derivatives of the numerator and denominator

Compute the derivatives of the numerator and denominator: $$(d/dx)(\ln x - 1) = \frac{1}{x}$$ $$(d/dx)(x-e) = 1$$ Therefore, our new limit expression is: $$\lim_{x \rightarrow e} \frac{\frac{1}{x}}{1}$$

Evaluate the limit

Plug in \(e\) for \(x\) in the simplified limit expression: $$\lim_{x \rightarrow e} \frac{\frac{1}{x}}{1} = \frac{1}{e}$$ So, the limit of the given function as x approaches e is \(\frac{1}{e}\).