The values of the constants `a , b` and `c` for which the function `f(x)={(1+a x)^(1//x)b ,x<0((x+c)^(1/3)-1)/((x+1)^(1/2)-1),x >0x=0` may be continuous at `x=0,` are `a=(log)_e(2/3),b=-2/3,c=1` `a="log")e` `(2/3),b2/3,c=-1` `a=(log)_e(2/3),b=2/3,c=1` (d) none of these

Answer

Option (C)
`a=log frac{2}{3}, b=frac{2}{3}, c=1`

Explanation

$$ a=\log \frac{2}{3}, b=\frac{2}{3}, c=1 $$
ATQ,


`f(x)=`\begin{cases}{l}(1+a x)^{\frac{1}{x}}, x<0 \\ b, x=0 \\ \frac{(x+c)^{\frac{1}{3}}-1}{(x+1)^{\frac{1}{2}}-1}, x>0\end{cases}$
If `f(x)` is continuous at `x=0` then
$$
\begin{aligned}
...