## Calculus AB and Calculus BC

## CHAPTER 2 Limits and Continuity

### C. THEOREMS ON LIMITS

If lim *f* (*x*) and lim *g*(*x*) are finite numbers, then:

(1) lim *kf* (*x*) = *k* lim *f* (*x*).

(2) lim[*f* (*x*) + *g*(*x*)] = lim *f* (*x*) + lim *g*(*x*).

(3) lim *f* (*x*)*g*(*x*) = (lim *f* (*x*))(lim *g*(*x*)).

(4) (if lim *g*(*x*) ≠ 0).

(5)

(6) THE SQUEEZE OR SANDWICH THEOREM. If *f* (*x*) ≤ *g*(*x*) ≤ *h*(*x*) and if

Figure N2–8 illustrates this theorem.

**FIGURE N2–8**

*Squeezing* function *g* between functions *f* and *h* forces *g* to have the same limit *L* at *x* = *c* as do *f* and *g.*

**EXAMPLE 9**

**EXAMPLE 10**

**EXAMPLE 11**

**EXAMPLE 12**

since, by the definition of in §A, *x* must be different from 3 as *x* → 3, the factor *x* − 3 may be removed *before* taking the limit.

**EXAMPLE 13**

**EXAMPLE 14**

the numerator approaches 1 while the denominator approaches 0; the limit does *not* exist.

**EXAMPLE 15**

**EXAMPLE 16**

**EXAMPLE 17**