Two differentiable functions and are defined on an open interval containing

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

Problem

Two differentiable functions

f

and

g

are defined on an open interval containing

x=3

. Their values near

x=3

are recorded below.

x	2.90	2.99	2.999	3.001	3.01	3.10
f(x)	−5.82	−5.982	−5.9982	−6.0018	−6.018	−6.18
g(x)	2.84	2.94	2.994	2.996	3.04	3.24

Assume that

f

and

g

remain finite and differentiable on the interval shown. (a) Estimate

\displaystyle\lim_{x\to 3}f(x)

and

\displaystyle\lim_{x\to 3}g(x)

from the data. (b) Using the tables, calculate an approximation for

\displaystyle\lim_{x\to 3}\dfrac{f(x)}{g(x)}

. (c) Indicate whether the expression

\dfrac{f(x)}{g(x)}

forms an indeterminate type as

x

approaches

3

, and briefly justify your answer using your estimates from part (a).