Question

Find the following limit.

Lim √(x + 2) - 3 / (x-7)
x→7

Which statements describe finding the limit shown?
Check all that apply.
A. Multiply by √x +2+3 / √x+2+3
B. Get x-1 in the numerator.
C. Get (x-7)(√x+2-3) in the denominator.
D. Divide out a common factor of x- 7.
E. Calculate the limit as 1/6

Answer 1

To find the limit as x approaches 7, we multiply by the conjugate of the numerator to rationalize it and then simplify, allowing us to cancel out common factors and compute the limit, which is 1/6.

Step-by-step explanation:

The task is to find the limit as x approaches 7 of the expression √(x + 2) - 3 / (x-7). To solve this problem, we must manipulate the expression to eliminate the indeterminate form that arises when directly substituting x = 7.

To do this, we implement the following steps:

• Multiply by the conjugate of the numerator, which is √(x + 2) + 3, both in the numerator and the denominator to rationalize the numerator.
• This results in (x + 2) - 9 in the numerator after simplifying, which simplifies further to x - 7.
• This allows us to divide out the common factor of (x - 7) from the numerator and denominator.
• Finally, we evaluate the new simplified expression at x = 7 to obtain the limit.

Calculating the limit yields the result of 1/6.