If `lim_(xrarr0)(2ax+(a-1)sinx)/(tan^3x)=l`,then `a+l` is equal to

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(2)/(9)`
D. `(4)/(9)`

Answer

D

Explanation

We have,
`lim_(xto0)(2ax+(a-1)sinx)/(tan^3x)=l`
`lim_(xto0)(2ax+(a-1)(sinx)/x)/(x^2((tanx)/x)^3)=l`
We observe that the numerator on LHS tends to `3a-1` and the denominator to `0`. So, the limit will be a finite number l only if
` 3a- 1 =0i.e., a=(1)/(3)`
` therefore lim_(xto0) ((2)/(3)(1-(sinx)/(x)))/(x^2((tanx)/(x))^3)=l`
`rArr lim_(xto0) ((x-sinx)/(x^3))/((tanx)/(x))^3=(3l)/(2)`
` rArr (1)/(6)=(3l)/(2)rArr l=(1)/(9) [because lim_(xto0) (x-sinx)/(x^3)=(1)/(6)]`
Hence, ` a+l=(1)/(3)+(1)/(0)=(4)/(9)`.