If `f(x)=root (3)(8x^(3)+mx^(2))-nx` such that `lim_(xrarroo)f(x)=1` then (A) `m+n=15` (B) `m-n=10` (C) `m-n=12` (D) `m+n=14`

Options

A. `m+n=15`
B. `m-n=10`
C. `m-n=12`
D. `m+n=14`

Answer

B, D

Explanation

`lim_(xrarroo)root(3)(8x^(3)+mx^(2))-nx`
`=lim_(xrarroo)((8-n^(3))x^(3)+mx^(2))/(root(3)((8x^(3)+mx^(2))^(2))+nx^(3)sqrt(8x^(3)+mx^(2))+n^(2)x^(2))`
for limit to be finite `8-n^(3)=0impliesn=2`
Hence
`lim_(xrarroo)f(x)=lim_(xrarroo)m/(root(3)((8+m/x)^(2))+2root(3)((8+m/x))+4)`
`=m/12=1impliesm=12`