E. OTHER BASIC LIMITS
E1. The basic trigonometric limit is:
if θ is measured in radians.
Example 22 __
Prove that .
SOLUTION: Since, for all x, −1 ≤ sin x ≤ 1, it follows that, if x > 0, then
. But as x → ∞, and 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 y 2 and y 3 approach 0 and that y 1
is squeezed between them.
Example 23 __
Find .
SOLUTION:.
Limit definition of e
E2. The number e can be defined as follows: