If `lim_(xto0) (1-cosx)/(e^(ax)-bx-1)=2` then find the values of a and b.

Answer

`a=1,b=1" and "a=-1,b=-1`

Explanation

`underset(xto0)lim(x^(2))/(e^(ax)-bx-1)=2`
`impliesunderset(xto0)lim(x^(2))/((1+(ax)/(1!)+(a^(2)x^(2))/(2!)+...)-bx-1)=2`
`impliesunderset(xto0)lim(x)/((a-b)+(a^(2)x)/(2!)+...)=2`
Limit exists if `a-b=0` or `a=b`
So, we have `underset(xto0)lim(x)/((a^(2)x)/(2))=2`
`:." "(2)/(a^(2))=2`
`implies" "a=+-1`
If `a=1," then " b=1" and "if a=-1," then "b=-1`