Consider three independent functions ,

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider three independent functions gg, hh, and kk defined on real numbers with the following properties. Function gg: • g(x)=x2−9x−3g(x)=\frac{x^2-9}{x-3} for x≠3x\neq 3; gg is undefined at x=3x=3. Function hh: • hh is defined for all real xx. • lim⁡x→1−h(x)=4\lim_{x\to 1^-}h(x)=4 and lim⁡x→1+h(x)=−2\lim_{x\to 1^+}h(x)=-2. • h(1)=7h(1)=7. Function kk: • k(x)=1(x+2)2k(x)=\frac{1}{(x+2)^2} for x≠−2x\neq -2; kk is undefined at x=−2x=-2. (1) Part (a): Evaluate and discuss the existence of the appropriate limits. (2) Part (b): Classify the type of discontinuity for each function with justification. (3) Part (c): Describe how to redefine the relevant function(s) to achieve continuity.

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