University:
California State University, Northridge
Course:
MATH 150AL | Calculus I Laboratory
Academic year:

2004
Views:

467

Pages:

18
Author:

azngangsta896gyt

Limits at infinity infinite value of variable Definition. L = lim f (x) x→∞ ⇔ ∀ε > 0 ∃N > 0 / x > N ⇒ |f (x) − L| < ε Definition. L = lim f (x) x→∞ ⇔ ∀ε > 0 ∃N > 0 / x > N ⇒ |f (x) − L| < ε 5x + 1 Example: lim =5 x→∞ x − 2 ⇔ ∀ε > 0 ∃N > 0 / x > N 5x + 1 ⇒ −5 <ε x−2 (in particular, | 5x+1 x−2 − 5| = 11 |x−2| < ε if N = 2 + 11 ε ) Properties: Let lim f (x) and lim g(x) exist, x→∞ x→∞ lim [f (x) + g(x)] = lim f (x) + lim g(x) x→∞ x→∞ x→∞ lim [f (x) − g(x)] = lim f (x) − lim g(x) x→∞ x→∞ x→∞ lim [f (x) · g(x)] = [ lim f (x)] · [ lim g(x)] x→∞ x→∞ x→∞ lim [f (x)/g(x)] = [ lim f (x)]/[ lim g(x)], x→∞ x→∞ x→∞ lim g(x) 6= 0 x→∞ q p n lim g(x) = n lim g(x), lim g(x) > 0, n even x→∞ x→∞ x→∞ EXAMPLES EXAMPLE 1. Evaluate limit 1 lim x→∞ x EXAMPLE 1. Evaluate limit 1 lim x→∞ x As variable x gets larger, 1/x gets smaller because 1 is being divided by a laaaaaaaarge number: 1 1 x = 1010, = 10 x 10 EXAMPLE 1. Evaluate limit 1 lim x→∞ x As variable x gets larger, 1/x gets smaller because 1 is being divided by a laaaaaaaarge number: 1 1 x = 1010, = 10 x 10 The limit is 0. 1 lim = 0. x→∞ x EXAMPLE 2. Limits 1 lim n = 0 x→∞ x 1 lim √ =0 n x→∞ x EXAMPLE 2. Limits 1 lim n = 0 x→∞ x 1 lim √ =0 n x→∞ x By the same argument 1 lim = 0, x→∞ x − 100 1 =0 lim √ n x→∞ x − 10000 EXAMPLE 3. Evaluate limit 1 lim √ x→∞ n x2 + x − C EXAMPLE 3. Evaluate limit 1 lim √ x→∞ n x2 + x − C Recall the graph of 2 y =x +x−C Observation: y= x 2+ x - C x→∞⇒y→∞ Value of C does not matter. The answer is 0. O -1/4-C EXAMPLE 4. Evaluate limit (not evoke graphs) 1 lim 2 x→∞ x − x − 1 EXAMPLE 4. Evaluate limit (not evoke graphs) 1 lim 2 x→∞ x − x − 1 Attention: indeterminacy ∞ − ∞ 1 = lim 1 1 x→∞ 2 x 1− − 2 x x 1 = lim 2 lim x→∞ x x→∞ 1 =0·1=0 1 1 1− − 2 x x EXAMPLE 5. Evaluate limit 5x3 − 2x2 − 1 lim x→∞ x3 − x + 1 EXAMPLE 5. Evaluate limit 5x3 − 2x2 − 1 lim x→∞ x3 − x + 1 Standard trick: divide by the highest degree of x 1 5x3 2x2 2 1 − − 5 − − 3 3 3 3 x x x x x = lim = lim 3 1 1 x→∞ x→∞ x x 1 1− 2+ 3 − + x x x3 x3 x3 5−0−0 = =5 1−0+0 EXAMPLE 6. Evaluate limit lim p x→∞ x−2 2x2 − x + 1 EXAMPLE 6. Evaluate limit lim p x→∞ x−2 2x2 − x + 1 Standard trick: divide by the highest degree of x x 2 x 2 − − 3 = lim √ x x = lim r x x x→∞ 2x2 − x + 1 x→∞ 2x2 x 1 √ − + x2 x2 x2 x2 1−0 1 =√ =√ . 2−0+0 2

Limits at Infinity. Infinite Value of Variable

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