Chapter 1: Limits
Section 1.4: Limits for Trig Functions
Example 1.4.1
Use the contents of Table 1.4.1 to evaluate limx→0sin3 x4 x.
Solution
The function sin3 x4 x is not defined at x=0 and no amount of factoring or other manipulation can eliminate the "division by zero" issue at x=0. Figure 1.4.1(a) and Table 1.4.1(a) give graphic and numeric evidence that the requisite limit is 3/4.
Figure 1.4.1(a) Graph of sin3 x4 x
Table 1.4.1(a) Table of values for sin3 x4 x
The requisite manipulations are given below. The first equality is valid because limx→ac fx=c limx→afx, for any nonzero constant c. To pass from the second to the third equality, reason that as x goes to zero, so also does 3 x. Then, to get the fourth equality, set θ=3 x.
limx→0sin3 x4 x
=34 limx→043 sin3 x4 x
=34 limx→0sin3 x3 x
=34 lim3 x→0sin3 x3 x
=34 limθ→0sinθθ
=34⋅1
=34
That Maple can obtain this same result is clear from the calculation limx→0sin3 x4 x = 34.
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