The answer is A. In this expression, if one attempts to evaluate the limit by substitution, it becomes apparent that there is an indeterminate form 0/0. Rationalizing the numerator will help to factor out and cancel, and thus remove, the algebraic expression that produces these zeroes. The limit then can be evaluated.
The expression in choice B, while it also contains a radical, is only defined for
x > 1. There is no function at all in the vicinity of x = -1, so the concept of the limit there is meaningless.
Expressions in choices C and D do not contain radicals, so the limits cannot be evaluated using rationalization.
To solve certain limit problems, you’ll need the Rationalization technique. When substitution doesn’t work in the original function when the answer comes in the form of `0/0 ` or `oo/oo` you can use Rationalization technique to manipulate the function until substitution does work.
Now coming to the problem,
1) The function has to be rationalized with (4+sqrt(18-x)) by multiplying and dividing. Thats the only way to solve the limits of this function. The complete procedure for the finding of the steps is given in the attachments below.
2) As the function is not in the form of `0/0` or `oo/oo ` when x= -1 is substituted, so no need of rationalization.
3) In this when x-> pi/2 the function is in the form `0/0 ` or `oo/oo`, This function can be rationalized by multiplying and dividing by (1+sinx)
`lim_(x-> pi/2) ((1-sinx)*(1+sinx))/((cosx)*(1+sinx))`
`lim_(x-> pi/2) ((1-sin^2(x)))/((cosx)*(1+sinx))`
`lim_(x-> pi/2) (cos^2 (x))/((cosx)*(1+sinx))`
`lim_(x-> pi/2) ((cosx)/(1+sinx))`
now on applying the value of x we get
`lim_(x-> pi/2) ((cosx)/(1+sinx)) = 0`
But this problem can also be done through the L 'Hospital rule so we can keep this aside
4)clearly the function can be done using the dividing out technique
Thought some of functions given in the options can also be solved by rationalization technique, but they have alternative methods to solve .
But, only option 1 is the only option which can be solved by the rationalization technique.