Nā nīnau hoʻohālike e kūkākūkā ana i nā palena o nā hana Algebraic
ʻO ka palena o kahi hana algebraic kahi manaʻo nui i ka calculus, e nānā ana i ke ʻano o kahi hana i ka wā e hoʻokokoke aku ai kona mau waiwai loli i kahi kiko. He mea nui ka hoʻomaopopo ʻana i nā palena i nā noi makemakika like ʻole, me ka nānā ʻana i ka makemakika a me ke kumu hoʻohālike. E wehewehe kēia ʻatikala i ke kumumanaʻo o ka palena o kahi hana algebraic ma ka hāʻawi ʻana i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā.
Manaʻo Kumu o nā Palena o nā Hana Algebraic
Ma mua o ko kākou komo ʻana i nā pilikia hoʻohālike, e nānā hou kākou i ke kumumanaʻo kumu o nā palena. ʻO ka palena o kahi hana \( f(x) \) i ka wā e hoʻokokoke aku ai ʻo \( x \) i ka waiwai \( a \) ua hōʻike ʻia e:
\[ \lim_{x \to a} f(x) = L \]
ʻo ia hoʻi, ua hoʻokokoke ka waiwai o \( f(x) \) iā \( L \) e like me ka hoʻokokoke ʻana o \( x \) iā \( a \).
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana Nīnau 1: Palena o nā Hana Algebraic Maʻalahi
E hoʻoholo i nā palena palena penei:
\[ \lim_{x \to 2} (3x + 4) \]
Kūkākūkā:
No kahi hana linear e like me kēia, hiki iā mākou ke pani pololei i ka waiwai o \( x \) me 2:
\[ \lim_{x \to 2} (3x + 4) = 3(2) + 4 = 6 + 4 = 10 \]
No laila, \( \lim_{x \to 2} (3x + 4) = 10 \).
Laʻana Nīnau 2: Palena o ka Hana Polynomial
E hoʻoholo i nā palena palena penei:
\[ \lim_{x \to -1} (x^2 + 2x + 1) \]
Kūkākūkā:
E like me ka nīnau mua, hiki iā mākou ke pani pololei i ka waiwai o \( x \) me -1 ma ka hana polynomial:
\[ \lim_{x \to -1} (x^2 + 2x + 1) = (-1)^2 + 2(-1) + 1 \]
\[ = 1 – 2 + 1 \]
\[ = 0 \]
No laila, \( \lim_{x \to -1} (x^2 + 2x + 1) = 0 \).
Laʻana Nīnau 3: Palena o nā Hana Algebraic me nā hapa
E hoʻoholo i nā palena palena penei:
\[ \lim_{x \to 3} \frac{x^2 – 9}{x – 3} \]
Kūkākūkā:
Inā mākou e pani pololei iā \( x = 3 \) i loko o ka hana, loaʻa iā mākou ke ʻano indeterminate \( \frac{0}{0} \). No ka hoʻoponopono ʻana i kēia, pono mākou e hoʻokaʻawale:
\[ \frac{x^2 – 9}{x – 3} = \frac{(x – 3)(x + 3)}{x – 3} \]
Ma mua o ka hoʻopau ʻana \( x – 3 \), e hoʻomaopopo ʻo \( x \neq 3 \), no laila hiki iā mākou ke hoʻopau \( x – 3 \):
\[ = x + 3 \]
I kēia manawa, e pani i ka \( x = 3 \):
\[ \lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 3 + 3 = 6 \]
No laila, \( \lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 6 \).
Laʻana Pilikia 4: Nā Palena o nā Hana me nā Aʻa
E hoʻoholo i nā palena palena penei:
\[ \lim_{x \to 4} \sqrt{2x + 1} \]
Kūkākūkā:
ʻOiai ʻo ka hana ma nā aʻa he hana hoʻomau, hiki iā mākou ke pani pololei i ka waiwai o \( x = 4 \):
\[ \lim_{x \to 4} \sqrt{2x + 1} = \sqrt{2(4) + 1} \]
\[ = \sqrt{8 + 1} \]
\[ = \sqrt{9} \]
\[ = 3 \]
No laila, \( \lim_{x \to 4} \sqrt{2x + 1} = 3 \).
Laʻana Nīnau 5: Palena o nā Hana Algebraic me ka Rationalization
E hoʻoholo i nā palena palena penei:
\[ \lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} \]
Kūkākūkā:
ʻO ka hoʻololi pololei \( x = 1 \) e loaʻa ai ke ʻano indeterminate \( \frac{0}{0} \). No laila pono mākou e hoʻomaopopo. E hoʻonui i ka numerator a me ka denominator me kā lākou mau hui like:
\[ \frac{\sqrt{x + 3} – 2}{x – 1} \times \frac{\sqrt{x + 3} + 2}{\sqrt{x + 3} + 2} = \frac{(\sqrt{x + 3})^2 – 2^2}{(x – 1)(\sqrt{x + 3} + 2)} \]
E hoʻomaʻalahi i ka helu:
\[ = \frac{x + 3 – 4}{(x – 1)(\sqrt{x + 3} + 2)} \]
\[ = \frac{x – 1}{(x – 1)(\sqrt{x + 3} + 2)} \]
Hoʻōki \( x – 1 \) (mai \( x \neq 1 \)):
\[ = \frac{1}{\sqrt{x + 3} + 2} \]
I kēia manawa, e pani i ka \( x = 1 \):
\[ \lim_{x \to 1} \frac{1}{\sqrt{x + 3} + 2} = \frac{1}{\sqrt{1 + 3} + 2} \]
\[ = \frac{1}{\sqrt{4} + 2} \]
\[ = \frac{1}{2 + 2} \]
\[ = \frac{1}{4} \]
No laila, \( \lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} = \frac{1}{4} \).
Ka hopena
ʻO ka hoʻomaopopo ʻana i nā palena o nā hana algebraic e pili ana i nā ʻano hana like ʻole e like me ka hoʻololi pololei, ka factorization, a me ka rationalization. Ma ka hoʻomaopopo ʻana i kēia mau ʻano hana, hiki iā mākou ke hoʻoponopono i nā ʻano pilikia palena like ʻole i ka calculus. I ka wā e kū ai i kahi hana indeterminate, e ʻimi mau i nā ala e hoʻomaʻalahi ai i ka hana i hiki ke helu pololei ʻia ka palena. Manaʻolana, ua kōkua nā pilikia hoʻohālike a me ke kūkākūkā ʻana ma luna iā ʻoe e hoʻomaopopo maikaʻi i kēia manaʻo.
