Choose the correct answer below. A. The numerator of the expression simplifies to for all x, so the limits are equal.
B. The limits and equal the same number when evaluated using direct substitution.
C. Since whenever x, it follows that the two expressions evaluate to the same number as x approaches .
D. Since each limit approaches , it follows that the limits are equal.

B. The limits Lim x-->7 (x²-9x+14)/x-7  and  lim x--> 7 (x-2) equals the same number when evaluated using direct substitution. The limit of both functions is 5

Step-by-step explanation:

Find the complete question in the attachment.

Given the limit of the functions

Lim x-->7 (x²-9x+14)/x-7

To solve this, we will need to factorize the quadratic function at the numerator first.

x²-9x+14

= x²-2x-7x+14

= x(x-2)-7(x-2)

= (x-7)(x-2)

The expression therefore becomes:

= lim x-->7 (x-7)(x-2)/x-7

= lim x-->7 (x-2)

Now substitute the value of x into the simplified function

lim x-->7 (x-2) = 7-2

lim x-->7 (x-2) = 5

Hence Lim x-->7 (x²-9x+14)/x-7 = 5

From the calculation above, it can be seen that Lim x-->7 (x²-9x+14)/x-7 = lim x--> 7 (x-2) = 5

Hence the correct answer based on the explanation above is B.

The limits Lim x-->7 (x²-9x+14)/x-7  and  lim x--> 7 (x-2) equals the same number when evaluated using direct substitution.