If f is differentiable at a, the limit lim_{x->a} [f(x) - f(a)]/(x - a) equals:

If f is differentiable at a, the limit lim_{x->a} [f(x) - f(a)]/(x - a) equals:

• A. f(a)
• B. f'(a)
• C. 0
• D. ∞

Answer: (B)

The derivative at a is defined as the limit of the difference quotient as x approaches a. Since f is differentiable at a, this limit exists and equals the derivative f′(a). In other words, lim_{x→a} [f(x) − f(a)]/(x − a) is exactly f′(a), the instantaneous rate of change or slope of the tangent to the graph at a. The function value f(a) is just the point on the graph, not the slope; the limit would be zero only if the derivative happens to be zero, and would be infinite only if the derivative didn’t exist or was unbounded, which contradicts differentiability.

The derivative at a is defined as the limit of the difference quotient as x approaches a. Since f is differentiable at a, this limit exists and equals the derivative f′(a). In other words, lim_{x→a} [f(x) − f(a)]/(x − a) is exactly f′(a), the instantaneous rate of change or slope of the tangent to the graph at a. The function value f(a) is just the point on the graph, not the slope; the limit would be zero only if the derivative happens to be zero, and would be infinite only if the derivative didn’t exist or was unbounded, which contradicts differentiability.