Question

Evaluate: \[ \lim_{x\to e}\frac{\log x-1}{x-e} \]

• A. \(1\)
• B. \(\frac12\)
• C. \(\frac1e\)
• D. Does not exist

Hint:

Whenever logarithmic limits produce the indeterminate form \(\frac00\), L'Hospital's Rule is usually the fastest method: \[ \frac{d}{dx}(\log x)=\frac1x \]

Answer 1

The Correct Option is C

Step 1: Check the form at $x=e$.
Top: $\log e-1=1-1=0$. Bottom: $e-e=0$. So we have $\dfrac00$, an indeterminate form.
Step 2: Recognise a derivative.
This limit is exactly the definition of the derivative of $\log x$ at $x=e$, because it has the shape $\dfrac{f(x)-f(e)}{x-e}$ with $f(x)=\log x$ (note $f(e)=1$).
Step 3: Or use L'Hospital's rule.
Since the form is $\dfrac00$, we may differentiate top and bottom separately.
Step 4: Differentiate the top.
$\dfrac{d}{dx}(\log x-1)=\dfrac1x$.
Step 5: Differentiate the bottom.
$\dfrac{d}{dx}(x-e)=1$.
Step 6: Take the limit.
\[ \lim_{x\to e}\frac{1/x}{1}=\frac1e. \] \[ \boxed{\frac1e} \]