Question

Which of the following is the correct limit definition for the derivative of g(x)=2x?

a) lim 1 h→0
b) lim 1 2ʰ-2ˣ /h h→0
c) lim 1 2ˣ⁺ʰ+2ˣ / h h→0
d) lim 1 2ˣ⁺ʰ -2ˣ /h h→ 0

Answer 1

The correct limit definition for the derivative of the function g(x) = 2x is (a) lim h→0 [2], since the h terms cancel out, leaving the derivative as a constant, which is 2.

Step-by-step explanation:

The limit definition for the derivative of a function g(x) is given by the expression:

lim h→σ_0 [(g(x+h) - g(x)) / h].

In the case of the function g(x) = 2x, we apply this definition:

1. First, we calculate the value of the function at x+h: g(x+h) = 2(x+h).
2. Next, we subtract the value of the function at x: 2(x+h) - 2x, which simplifies to 2h.
3. Now, we divide by h and take the limit as h approaches zero: lim h→σ_0 [2h / h].
4. Finally, this simplifies to lim h→σ_0 [2] because the h cancels out in the numerator and denominator, which simply equals 2 since it does not depend on h.

Therefore, the correct limit definition for the derivative of g(x) = 2x is:

lim h→σ_0 [2].

Among the options given by the student, the correct option is (a) which represents this result.