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) Image (if lim g(x) ≠ 0).

(5) Image

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

Figure N2–8 illustrates this theorem.

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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

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EXAMPLE 10

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EXAMPLE 11

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EXAMPLE 12

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

EXAMPLE 13

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EXAMPLE 14

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

EXAMPLE 15

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EXAMPLE 16

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EXAMPLE 17

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