Define the function k(x)= x^ 2 -9 x-3 + ( 50x )(x-3)

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

Problem

Define the function k(x)=x2−9x−3+(sin⁡50x)(x−3).k(x)=\frac{x^{2}-9}{x-3}+\bigl(\sin 50x\bigr)(x-3). A student plots kk on the standard viewing window −10≤x≤10-10 \le x \le 10, −10≤y≤10-10 \le y \le 10. The graph appears to pass smoothly through the point (3,6)(3,6) with no visible irregularities. Investigate the limit lim⁡x→3k(x)\displaystyle \lim_{x\to 3} k(x) and comment on how the graph’s appearance might mislead someone who relies only on the default window. (a) Calculate lim⁡x→3k(x)\displaystyle \lim_{x\to 3} k(x) algebraically. Show all necessary steps. (b) The term (sin⁡50x)(x−3)(\sin 50x)(x-3) is too small to notice in the default window. Describe quantitatively why this term does not affect the plotted point at (3,6)(3,6), yet could become visible with an appropriate zoom. (c) Propose a specific viewing window (state numerical bounds for both axes) that would reveal the effect of the oscillatory term near x=3x=3, and explain how this window would help confirm the analytical limit from part (a).

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