Final answer:
The limit lim x→c (f(x)−f(c))/(x-c) represents the definition of the derivative of the function f at x = c and has the indeterminate form 0/0. If f'(c) exists, the limit resolves to the derivative f'(c).
Step-by-step explanation:
The question explores the concept of limits in calculus, specifically the limit of a function as it approaches a particular point. The given limit, lim x→c (f(x)−f(c))/(x-c), can be recognized as the definition of the derivative of f at x = c, assuming f'(c) exists. When we apply the limit, both the numerator and denominator approach zero, resulting in the indeterminate form 0/0. By applying L'Hôpital's Rule or knowing that the limit is the definition of the derivative, we can conclude that the limit resolves to f'(c), provided the derivative exists at that point.
The concept of limits is crucial in various mathematics and physics applications, such as finding the slope of a tangent line to a curve at a point, understanding behaviour of functions near asymptotes, or in calculating instantaneous rates of change. Asymptotes, for example, as mentioned in the reference provided, indicate values that a function approaches but never reaches, creating a boundary of behaviour for the function. In the context of probability, the principle of limits helps us understand that the probability of a continuous random variable at a single point is zero, mirroring the infinitesimal nature of differential calculus.