Barrons AP Calculus

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: