Final answer:
To find the limit when faced with a difference of squares in the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which after expansion helps to simplify the expression and allows us to find the limit. Option (b) correctly describes this process; the final value of the limit depends on additional context not provided.
Step-by-step explanation:
The question references the process of finding a limit where we encounter a difference of squares in the denominator that prevents direct substitution due to a zero in the denominator. To solve this, we multiply both the numerator and the denominator by the conjugate of the denominator. This technique is useful in limit problems with radicals.
Let's say our initial limit expression is of the form:
( lim_{x to a} frac{x - 1}{sqrt{x + 2} - 3} )
To find this limit, we would multiply by the conjugate:
( frac{sqrt{x + 2} + 3}sqrt{x + 2} + 3} )
Upon expansion, the denominator becomes:
( (sqrt{x + 2} - 3)(sqrt{x + 2} + 3) = x - 7 )
This eliminates the radical and creates a quadratic expression which we can further simplify if it has common factors with the numerator. The correct option that describes finding the limit is (b) Get * (x - 7) * (sqrt{x + 2} - 3) in the denominator.
The other steps may involve further simplification and eventually, after dividing out any common factors (if applicable), we calculate the final value of the limit. The actual limit, however, depends on the specific values of x and the function. Without additional context, we are not given enough information to definitively state that the limit is 1/6, so option (d) cannot be verified here.
In summary, when faced with a rational function that has a non-removable discontinuity at the point of interest, we often use the multiplication by the conjugate strategy to simplify the expression and find the limit.