Final answer:
To solve the limit, we multiply by the conjugate of the numerator to rationalize it. We simplify the expression to find that the limit is 1/6, confirming that options 'a', 'c', 'd', and 'e' are correct in describing the process of finding the given limit.
Step-by-step explanation:
When dealing with the limit ℓₓ→7 √ₓ+2-3/ₓ-7, we encounter an indeterminate form of type 0/0. To resolve this, we can use the technique of rationalizing the numerator. Specifically, we multiply the given expression by the conjugate of the numerator over itself to create a difference of squares in the numerator which can help to cancel out the (ₓ-7) term in the denominator.
Multiplying the expression by the conjugate of the numerator √ₓ+2+3/√ₓ+2+3 is an appropriate step. This process will not immediately lead to an (ₓ-1) in the numerator as originally suggested in option 'b', but it will change the form of the expression and allow for further simplification. The resulting expression after this rationalization step would be:
((ₓ+2) - 9)/((ₓ-7)(√ₓ+2+3))
After simplifying and combining like terms in the numerator, we will have (ₓ-7) in the denominator, directly addressing option 'c'. At this point, we can divide out the common factor of (ₓ-7) as stated in option 'd'. Calculating the limit now becomes manageable, and upon further simplification, we eventually find that the limit is ⅖ which confirms option 'e' to be correct. The final step is always to check the answer to see if it is reasonable, in line with the guiding principles of algebraic simplification.