## 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.

**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:

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*.