Calculus AB and Calculus BC

CHAPTER 2 Limits and Continuity

E. OTHER BASIC LIMITS

E1. The basic trigonometric limit is:

Image if θ is measured in radians.

EXAMPLE 22

Prove that Image

SOLUTION: Since, for all x, −1 ≤ sin x ≤ 1, it follows that, if x > 0, then Image But as x → ∞, Image both approach 0; therefore by the Squeeze theorem, Image must also approach 0. To obtain graphical confirmation of this fact, and of the additional fact that Image also equals 0, graph

Image

in [−4π, 4π] × [−1, 1]. Observe, as x → ±∞, that y2 and y3, approach 0 and that y1 is squeezed between them.

EXAMPLE 23

Find Image

SOLUTION: Image

Limit definition of e

E2. The number e can be defined as follows:

Image

The value of e can be approximated on a graphing calculator to a large number of decimal places by evaluating

Image

for large values of x.