Consider the function f(x)= (x^2-1)(x-4) (x-1)(x^2-4x+3)
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Consider the function
f(x)=(x−1)(x2−4x+3)(x2−1)(x−4).
The numerator and denominator are given in completely factored form.
(1) Determine the complete set of all real numbers at which f is not continuous. Present your answer as a union of distinct points.
(2) Classify each discontinuity found in part (a) as removable, jump, or infinite. Justify every classification by evaluating the necessary one-sided and/or two-sided limits.
(3) Rewrite f as a new function h obtained by "removing" every removable discontinuity while leaving the domain elsewhere unchanged. Give an explicit formula for h defined on its domain.
(4) Prove that h is continuous at every x in its domain. Your proof should make direct reference to the formal definition of continuity and to your work in parts (a)–(c).