If f is differentiable at a, what is lim_{x->a} [f(x) - f(a)] / (x - a)?

If f is differentiable at a, what is lim_{x->a} [f(x) - f(a)] / (x - a)?

• A. 0
• B. f(a)
• C. f'(a)
• D. ∞

Answer: (C)

Differentiability at a means the slope of the secant line between (a, f(a)) and (x, f(x)) approaches a single value as x → a. That limiting slope is exactly the derivative at a. So, when f is differentiable at a, the limit of [f(x) - f(a)]/(x - a) as x → a exists and equals f'(a). This is the intuition behind the derivative: near a, the function behaves like a straight line with slope f'(a). This limit isn’t necessarily zero; it’s the actual rate of change of f at a, which is f'(a). It isn’t f(a) because f(a) is a function value, not a slope, and it isn’t infinite because differentiability ensures a finite tangent slope. As a check, with a simple example like f(x) = x^2 at a = 3, the limit equals f'(3) = 6.

Differentiability at a means the slope of the secant line between (a, f(a)) and (x, f(x)) approaches a single value as x → a. That limiting slope is exactly the derivative at a. So, when f is differentiable at a, the limit of [f(x) - f(a)]/(x - a) as x → a exists and equals f'(a). This is the intuition behind the derivative: near a, the function behaves like a straight line with slope f'(a).

This limit isn’t necessarily zero; it’s the actual rate of change of f at a, which is f'(a). It isn’t f(a) because f(a) is a function value, not a slope, and it isn’t infinite because differentiability ensures a finite tangent slope. As a check, with a simple example like f(x) = x^2 at a = 3, the limit equals f'(3) = 6.