Final answer:
The limit lim₍ₓ→f(1)-f(x)/(x-1) where f(x) = e^x, as x approaches 1, is found by differentiating the numerator and the denominator. The result is e, so the correct answer is option c.
Step-by-step explanation:
The given problem is to evaluate the limit of the function f(x) = e^x as x approaches 1. The limit expression is lim₍ₓ→f(1)-f(x)/(x-1). This is a standard limit that tests the derivative of the function f(x) at x = 1.
To solve this, we apply L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a certain value results in an indeterminate form like 0/0 or ∞/∞, then the limit can be found as the limit of the derivatives of the two functions, provided they exist. Thus, we differentiate the numerator and the denominator with respect to x.
The derivative of f(x) is f'(x) = e^x and the derivative of g(x) = x - 1 is g'(x) = 1. Substituting x = 1 into f'(x) gives us e. Therefore, the limit is simply e, which corresponds to option c.