Assume that `lim_(thetararr-1) f(theta)` exists and `(theta^(2)+theta-2)/(theta+3)le(f(theta))/(theta^(2))le(theta^(2)+2theta-1)/(theta+3)` holds for certain interval containing the point `theta=-1 " then "lim_(thetararr-1) f(theta)`

Options

A. is equal to `f(-1)`
B. is equal to 1
C. is non-existent
D. is equal to `-1`

Answer

A, D

Explanation

`(theta^(2)+theta-2)/(theta+3)le(f(theta))/(theta^(2))le(theta^(2)+2theta-1)/(theta+3)`
`"Putting "theta=-1, " we get"`
`(1-1-2)/(2)lef(-1)le(1-2-1)/(2)`
`rArr -1 le f(-1)le -1 " "rArr" "f(-1)=-1`
Also, `underset(thetararr-1)(lim)(theta^(2)+theta-2)/(theta+3)=-1=underset(thetararr-1)(lim)(theta^(2)+2theta-1)/(theta+3)`
Using sandwich theorem
`underset(theta rarr-1)(lim)(f(theta))/(theta^(2))=-1,underset(thetararr-1)(lim)f(theta)=-1`