Consider the function defined for all real numbers.
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the function f(x)=3x defined for all real numbers.
(a) Prove that f is continuous at x=0 by evaluating x→0limf(x) and comparing it with f(0). Clearly state any theorems you invoke.
(b) Using the limit definition of the derivative, write the difference quotient for f at x=0 and determine whether the limit exists. Show every algebraic step in your reasoning.
(c) Interpret the result from part (b) in terms of the slope of the tangent line at x=0. Describe what feature the graph exhibits at this point and explain how it exemplifies the relationship between continuity and differentiability.