Tables for two functions and are given for values near

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

Problem

Tables for two functions rr and ss are given for xx values near 11.
x0.900.950.991.011.051.10
r(x)2.812.90252.98013.02013.10253.21
x0.900.950.991.011.051.10
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s(x)0.3140.1560.010−0.010−0.156−0.314
Consider the ratio r(x)−3s(x)\displaystyle\dfrac{r(x)-3}{s(x)}. (a) Using the data, estimate lim⁡x→1(r(x)−3)\displaystyle\lim_{x\to 1}\bigl(r(x)-3\bigr) and lim⁡x→1s(x)\displaystyle\lim_{x\to 1}s(x). (b) Compute an approximation for lim⁡x→1r(x)−3s(x)\displaystyle\lim_{x\to 1}\dfrac{r(x)-3}{s(x)}, showing numerical evidence to support your answer. (c) Explain the significance of your result in part (b) with respect to the relative rates at which r(x)r(x) approaches 33 and s(x)s(x) approaches 00. Provide a brief verbal interpretation.

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