Barrons AP Calculus - David Bock

(ii) when then the graph of has no horizontal asymptotes;

(iii) when is a horizontal asymptote of the graph of

EXAMPLE 21

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

→ ∞, 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: