Question

$f\left(x\right)=\frac{p+q^{\frac{1}{x}}}{r+s^{\frac{1}{x}}},s<1,q<1,r\ne 0,f\left(0\right)=1$, is left continuous at $x=0$ then

• A. $p=0$
• B. $p=r$
• C. $p=q$
• D. $p\ne q$

Answer 1

The correct option is B $p=r$

$f\left(x\right)=\frac{p+q^{1/x}}{r+s^{1/x}}$

If $f\left(x\right)$ is right continuous

$\Rightarrow \underset{x\to 0^{+}}{lim}f\left(x\right)=f\left(0\right)=1$

$\Rightarrow \underset{x\to 0^{+}}{lim}\frac{p+q^{1/x}}{r+s^{1/x}}=1$

as $x\to 0^{+};\frac{1}{x}\to +\infty$

Given $s<1$

$\Rightarrow s=\frac{1}{a},a>1$

Similarly, $q=\frac{1}{b},b>1$

$=\underset{x\to 0^{+}}{lim}\frac{p+{\left(\frac{1}{b}\right)}^{\frac{1}{x}}}{r+{\left(\frac{1}{a}\right)}^{\frac{1}{x}}}$

$=\frac{p+\frac{1}{b^{\infty }}}{r+\frac{1}{a^{\infty }}}=\frac{p+0}{r+0}=\frac{p}{r}=1$

$\Rightarrow p=r$

Question should be changed to right continuous instead of left continuous.