Question

If $f\left(x\right)=x^{\frac{1}{x-1}}$ for $x\ne 1$ and $f$ is continuous at $x=1$ then $f\left(1\right)=$

• A. $e$
• B. $e^{-1}$
• C. $e^{-2}$
• D. $e^2$

Answer 1

Answer: (A)

$e$

$f\left(1\right)=RHLatx=1={lim}_{h\to 0}f(1+h)={lim}_{h\to 0}(1+h{)}^{\left(\frac{1}{1+h-1}\right)}={lim}_{h\to 0}(1+h{)}^{\left(\frac{1}{h}\right)}=e^{{lim}_{h\to 0}\left(\frac{1}{h}\right)(1+h-1)}=e$