Calculus AB and Calculus BC
CHAPTER 2 Limits and Continuity
E. OTHER BASIC LIMITS
E1. The basic trigonometric limit is:
if θ is measured in radians.
SOLUTION: Since, for all x, −1 ≤ sin x ≤ 1, it follows that, if x > 0, then But as x → ∞, both approach 0; therefore by the Squeeze theorem, must also approach 0. To obtain graphical confirmation of this fact, and of the additional fact that also equals 0, graph
in [−4π, 4π] × [−1, 1]. Observe, as x → ±∞, that y2 and y3, approach 0 and that y1 is squeezed between them.
Limit definition of e
E2. The number e can be defined as follows:
The value of e can be approximated on a graphing calculator to a large number of decimal places by evaluating
for large values of x.