A function is defined on an open interval containing except possibly at

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

Problem

A function hh is defined on an open interval containing x=2x=2 except possibly at x=2x=2. A numerical investigation shows the following: when xx is 2.001, h(x)=3.01h(x)=3.01; when xx is 2.0001, h(x)=3.001h(x)=3.001; when xx is 1.999, h(x)=4.97h(x)=4.97; and when xx is 1.9999, h(x)=4.999h(x)=4.999. No other information about hh is provided. Answer the questions that follow, using precise limit notation and complete sentences where requested. (a) State, in words, the condition that must be satisfied by the one-sided limits of hh at x=2x=2 for the two-sided limit lim⁡x→2h(x)\displaystyle \lim_{x\to2}h(x) to exist. (b) Based on the numerical evidence, write an appropriate statement for lim⁡x→2−h(x)\displaystyle \lim_{x\to2^-}h(x) and another for lim⁡x→2+h(x)\displaystyle \lim_{x\to2^+}h(x). Explain how you arrived at each value. (c) Determine whether lim⁡x→2h(x)\displaystyle \lim_{x\to2}h(x) exists. Justify your conclusion in the context of part (a).

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