Final answer:
The limit as h approaches 0 of the function (f(3 + h) - f(3)) / h, when f(x) = x², is computed using the difference quotient method and results in the answer of 6. This represents the derivative of the function at x = 3.
Step-by-step explanation:
The question asks us to compute the limit of the function f(x) = x² as h approaches 0 using the difference quotient method, which is a fundamental concept in calculus used to find the derivative of a function. This specific question is looking at the function value at 3 + h and at 3, and then dividing the difference by h. This is expressed as:
lim(h→0) [(f(3 + h) - f(3)) / h]
Firstly, we need to work out the expression f(3 + h) which is:
((3 + h)²) = 9 + 6h + h²
Subtracting f(3), which is 9, from the above expression we get:
(9 + 6h + h²) - 9 = 6h + h²
Dividing by h gives us:
(6h + h²) / h = 6 + h
Now, we apply the limit as h approaches 0:
lim(h→0) (6 + h) = 6
Therefore, the derivative of f(x) at x = 3 is 6. Hence, the multiple choice answer (MCQ answer) for the limit as h approaches 0 of (f(3 + h) - f(3)) / h, when f(x) = x², is (d) 6.1