154
Chapter 2 | Limits
2.2 EXERCISES For the following exercises, consider the function f ( x ) = x 2 −1 | x −1| . 30. [T] Complete the following table for the function. Round your solutions to four decimal places. x f ( x ) x f ( x )
table to evaluate the limits. Round your solutions to eight decimal places. 35. [T] lim x →0 sin2 x x ; ±0.1, ±0.01, ±0.001, ±.0001 x sin2 x x x sin2 x x
−0.1
a.
0.1
e.
0.9
a.
1.1
e.
−0.01
b.
0.01
f.
0.99
b.
1.01
f.
−0.001
c.
0.001
g.
0.999
c.
1.001
g.
−0.0001 d.
0.0001 h.
0.9999 d.
1.0001 h.
sin3 x x
36. [T] lim x →0
±0.1, ±0.01, ±0.001, ±0.0001
31. What do your results in the preceding exercise indicate about the two-sided limit lim x →1 f ( x )? Explain your response. For the following exercises, consider the function f ( x ) = (1+ x ) 1/ x . 32. [T] Make a table showing the values of f for x = −0.01, −0.001, −0.0001, −0.00001 and for x = 0.01, 0.001, 0.0001, 0.00001. Round your solutions to five decimal places. x f ( x ) x f ( x )
sin3 x x
sin3 x x
X
x
−0.1
a.
0.1
e.
−0.01
b.
0.01
f.
−0.001
c.
0.001
g.
−0.0001 d.
0.0001 h.
−0.01
a.
0.01
e.
37. Use the preceding two exercises to conjecture (guess) the value of the following limit: lim x →0 sin ax x for a , a positive real value. [T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.
−0.001
b.
0.001
f.
−0.0001
c.
0.0001
g.
−0.00001 d.
0.00001 h.
33. What does the table of values in the preceding exercise indicate about the function f ( x ) = (1+ x ) 1/ x ? 34. To which mathematical constant does the limit in the preceding exercise appear to be getting closer? In the following exercises, use the given values to set up a
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online