Question

Value of lim h → 0 [√(9 + h) - 3] / h.

A) 1/6
B) 1/3
C) 1/2
D) 2/3

Answer 1

The limit as h approaches 0 of [\u221A(9 + h) - 3] / h is found by rationalizing the numerator and then simplifying the expression, which yields a value of 1/6.

Step-by-step explanation:

The question is asking for the value of the limit as h approaches 0 of the expression [\u221A(9 + h) - 3] / h. To solve this, we can utilize the conjugate of the numerator to rationalize it. By multiplying both the numerator and the denominator by the conjugate \u221A(9 + h) + 3, we can simplify the expression and find the limit without the direct substitution, which in this case would result in an indeterminate form 0/0.

Here is the step-by-step simplification process:

1. Multiply both the numerator and the denominator by the conjugate to obtain:
   
   (\u221A(9 + h) - 3)(\u221A(9 + h) + 3) / (h(\u221A(9 + h) + 3)).
2. Simplify the numerator using the difference of squares formula:
   
   (9 + h) - 9 = h.
3. Now the expression looks like h / (h(\u221A(9 + h) + 3)).
4. Cancel out the h terms in the numerator and the denominator.
5. The simplified expression is now 1 / (\u221A(9 + h) + 3).
6. Finally, take the limit as h approaches 0 to get 1 / (\u221A(9) + 3) = 1 / 6.

Therefore, the value of the limit is 1/6, which corresponds to option A.