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 exact value of the limit is \(\frac{1}{e}\).

Step-by-step solution

Recall the definition of the derivate

Recall that the definition of the derivative of a function f(x) at a point x=c is given by: $$f'(c)=\lim_{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$$

Identify the function and the point

Identify the function and the point involved in the given limit. We can rewrite the given limit as : $$\lim_{x \rightarrow e} \frac{\ln x-\ln e}{x-e}$$ In this case, the function is \(f(x) = \ln x\) and the point is \(x=e\). We have \(f(e) = \ln e = 1\).

Connect the limit with the derivative definition

Now, we can see that the given limit matches the form of the definition of the derivative: $$\lim_{x \rightarrow e} \frac{f(x)-f(e)}{x-e} = f'(e)$$ So, we are actually trying to find the derivative of the function \(f(x) = \ln x\) at the point \(x=e\), i.e., \(f'(e)\).

Find the derivative of the function

To find the derivative of \(f(x) = \ln x\), we need to know the derivative of the natural logarithm function. The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\). So, the derivative of the function is \(f'(x) = \frac{1}{x}\).

Evaluate the derivative at the point x=e

Now, we need to evaluate the derivative of the function at the point \(x=e\). Substitute \(x=e\) into the derivative: $$f'(e) = \frac{1}{e}$$

Conclusion

So, using the definition of the derivative, we have found that the given limit is equal to the derivative of the function \(f(x) = \ln x\) at the point \(x=e\). The exact value of the limit is: $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e} = \frac{1}{e}$$