Oblicz granice:- Zadanie 2.32: Matematyka 3. Zakres rozszerzony. Po gimnazjum

Rozwiązanie

$a) x \to + \infty lim [(1 - x)^{2} (x + 3)] = x \to + \infty lim [(1 - 2 x + x^{2}) (x + 3)] =$

$= x \to + \infty lim (x + 3 - 2 x^{2} - 6 x + x^{3} + 3 x^{2}) = x \to + \infty lim (x^{3} + x^{2} - 5 x + 3) =$

$= x \to + \infty lim x^{3} (1 + \frac{1}{x} - \frac{5}{x ^{2}} + \frac{3}{x ^{3}}) = + \infty \cdot 1 = + \infty$

$b) x \to - \infty lim [- 3 x (2 - x) (2 + x)] = x \to - \infty lim [- 3 x (4 - x^{2})] =$

$= x \to - \infty lim (- 12 x + 3 x^{3}) = x \to - \infty lim x^{3} (3 - \frac{12}{x ^{2}}) = - \infty \cdot 3 = - \infty$

$c) x \to + \infty lim [(x - 3)^{3} (2 - 3 x)^{2}] = x \to + \infty lim [(x^{3} - 3 \cdot x^{2} \cdot 3 + 3 \cdot x \cdot 3^{2} - 3^{3}) (4 - 12 x + 9 x^{2})] =$

$= x \to + \infty lim [(x^{3} - 9 x^{2} + 27 x - 27) (4 - 12 x + 9 x^{2})] =$

$= x \to + \infty lim (4 x^{3} - 12 x^{4} + 9 x^{5} - 36 x^{2} + 108 x^{3} - 81 x^{4} + 108 x - 324 x^{2} + 243 x^{3} - 108 + 324 x - 243 x^{2}) =$

$= x \to + \infty lim (9 x^{5} - 93 x^{4} + 355 x^{3} - 603 x^{2} + 432 x - 108) =$

$= x \to + \infty lim x^{5} (9 - \frac{93}{x} + \frac{355}{x ^{2}} - \frac{603}{x ^{3}} + \frac{432}{x ^{4}} - \frac{108}{x ^{5}}) = + \infty \cdot 9 = + \infty$

$d) x \to - \infty lim [(5 - x) (3 - x)^{2} (x + 2)] = x \to - \infty lim [(5 - x) (x + 2) (9 - 6 x + x^{2})] =$

$= x \to - \infty lim [(5 x + 10 - x^{2} - 2 x) (9 - 6 x + x^{2})] =$

$= x \to - \infty lim (45 x - 30 x^{2} + 5 x^{3} + 90 - 60 x + 10 x^{2} - 9 x^{2} + 6 x^{3} - x^{4} - 18 x + 12 x^{2} - 2 x^{3}) =$

$= x \to - \infty lim (- x^{4} + 9 x^{3} - 17 x^{2} - 33 x + 90) =$

$= x \to - \infty lim x^{4} (- 1 + \frac{9}{x} - \frac{17}{x ^{2}} - \frac{33}{x ^{3}} + \frac{90}{x ^{4}}) = + \infty \cdot (- 1) = - \infty$

$e) x \to + \infty lim [(1 - x) (1 + x)] = x \to + \infty lim [1^{2} - x^{2}] =$

$= x \to + \infty lim (1 - x) = x \to + \infty lim x \cdot (\frac{1}{x} - 1) = + \infty \cdot (- 1) = - \infty$

$f) x \to - \infty lim [(9 - 2 x)^{3} (7 - x)^{3}] =$

$= x \to - \infty lim [(9^{3} - 3 \cdot 9^{2} \cdot 2 x + 3 \cdot 9 \cdot (2 x)^{2} - (2 x)^{3}) (7^{3} - 3 \cdot 7^{2} \cdot x + 3 \cdot 7 \cdot x^{2} - x^{3})] =$

$= x \to - \infty lim [(729 - 486 x + 108 x^{2} - 8 x^{3}) (343 - 147 x + 21 x^{2} - x^{3})] =$

$= x \to - \infty lim [x^{3} \cdot (\frac{729}{x ^{3}} - \frac{486}{x ^{2}} + \frac{108}{x} - 8) \cdot x^{3} (\frac{343}{x ^{3}} - \frac{147}{x ^{2}} + \frac{21}{x} - 1)] =$

$= x \to - \infty lim x^{6} (\frac{729}{x ^{3}} - \frac{486}{x ^{2}} + \frac{108}{x} - 8) (\frac{343}{x ^{3}} - \frac{147}{x ^{2}} + \frac{21}{x} - 1) = + \infty \cdot (- 8) \cdot (- 1) = + \infty$

Magdalena Matusik

Nauczycielka matematyki

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