Nā nīnau hoʻohālike a me ke kūkākūkā ʻana i nā waiwai o nā palena hana
Pendahuluan
ʻO ka palena o kahi hana he manaʻo nui ia i ka calculus e pāʻani ana i kahi kuleana koʻikoʻi i ka loiloi makemakika a me nā noi ʻepekema like ʻole. Kōkua nā palena hana iā mākou e hoʻomaopopo i ke ʻano o kahi hana i ka wā e hoʻokokoke aku ai kahi loli i kahi waiwai. Hāʻawi kekahi mau waiwai o nā palena hana i nā mea hana no ka helu ʻana a me ka hoʻoponopono ʻana i nā palena me ka maʻalahi. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau pilikia hoʻohālike a kūkākūkā i nā waiwai o nā palena hana.
Nā Waiwai o nā Palena Hana
Ma mua o ko mākou komo ʻana i nā pilikia hoʻohālike, e nānā hou kākou i kekahi mau waiwai kumu o nā palena hana i hoʻohana pinepine ʻia:
1. Palena o ka Hoʻohui
\[
\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
\]
2. Palena Hoʻonui
\[
\lim_{x \to a}[f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)
\]
3. Palena Hoʻolaha
\[
\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{i hāʻawi ʻia } \lim_{x \to a} g(x) \neq 0
\]
4. Palena Unahi Mau
\[
\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)
\]
5. Palena ʻIke
\[
lim_{x `to a} x = a
\]
6. Palena o ka Hana Mau
\[
\lim_{x \to a} c = c, \quad \text{kahi he mau ʻo c}
\]
Me ka hoʻomaopopo ʻana i kēia mau waiwai maʻamau, e hoʻopili kākou iā lākou i kekahi mau pilikia hoʻohālike.
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana Nīnau 1
E hāʻawi i nā hopena o:
\[
lim_{x 3} (2x^2 + 5x – 1)
\]
Kūkākūkā:
No ka hoʻoponopono ʻana i kēia palena, hiki iā mākou ke hoʻokomo pololei i ka waiwai x = 3 i loko o ka hana no ka mea he polynomial kēia hana a ua hoʻomau nā polynomials ma ko lākou ʻāpana.
\[
lim_{x 3} (2x^2 + 5x – 1) = 2(3)^2 + 5(3) – 1
\]
E helu i kēlā me kēia ʻanuʻu:
\[
= 2(9) + 15 – 1 = 18 + 15 – 1 = 32
\]
No laila:
\[
lim_{x 3} (2x^2 + 5x – 1) = 32
\]
Laʻana Nīnau 2
Helu:
\[
\lim_{x \to -2} \frac{3x^3 + 4x + 2}{x + 2}
\]
Kūkākūkā:
Ma kēia laʻana, ʻo ka hoʻokomo pololei ʻana iā x = -2 i loko o ke ʻano hapa e hana i ke ʻano indeterminate \( \frac{0}{0} \), no laila pono mākou e helu iā ia ma kekahi ʻano ʻē aʻe. ʻO kekahi ʻano hana ma ka hoʻohālikelike ʻana i ka helu.
E hoʻohālikelike i ka helu \( 3x^3 + 4x + 2 \):
Ma ka hoʻāʻo ʻana i ka waiwai o \( x = -2 \) ma ke koena o ka mahele, loaʻa iā mākou:
\[
3(-2)^3 + 4(-2) + 2 = -24 – 8 + 2 = -30 \quad \text{(no laila, ʻaʻole hiki ke hoʻonui hou ʻia kēia me ke kōkua ʻole o nā ʻano hana ʻē aʻe)}
\]
Hōʻike kēia he mea kūpono ʻole paha ke ʻano hana factorization pololei. A i ʻole, hiki iā mākou ke hoʻāʻo i ke ʻano hana a L'Hôpital. Inā mākou e hoʻokaʻawale i ka numerator a me ka denominator:
Mea helu: Hoʻokaʻawale ʻo \( 3x^3 + 4x + 2 \) i \( 9x^2 + 4 \).
ʻO ka Denominator: \( x + 2 \) e hoʻokaʻawale i \( 1 \).
A laila e hoʻopili iā L'Hôpital:
\[
lim_{x \to -2} \frac{9x^2 + 4}{1} = 9(-2)^2 + 4 = 9(4) + 4 = 36 + 4 = 40
\]
No laila:
\[
lim_{x _to -2} 3x^3 + 4x + 2}{x + 2} = 40
\]
Laʻana Nīnau 3
E huli:
\[
lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4}
\]
Kūkākūkā:
No nā pilikia palena i ka wā \( x \to \infty \), hiki iā mākou ke puʻunaue i kēlā me kēia ʻāpana ma ke kekelē kiʻekiʻe loa o x ma ka denominator, ʻo ia hoʻi \( x^2 \).
\[
\lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4} = \lim_{x \to \infty} \frac{5 – \frac{2}{x} + \frac{3}{x^2}}{1 + \frac{4}{x^2}}
\]
No ka mea, i ka wā \( x \to \infty \), \( \frac{1}{x} \to 0 \) a me \( \frac{1}{x^2} \to 0 \), a laila:
\[
\lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4} = \frac{5 – 0 + 0}{1 + 0} = 5
\]
No laila,
\[
lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4} = 5
\]
Laʻana Nīnau 4
E hāʻawi i nā hopena o:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x}
\]
Kūkākūkā:
Ua ʻike mākou mai nā ʻano o nā palena:
\[
lim_{x 0} \frac{\sin(x)}{x} = 1
\]
I kēia manawa, hoʻololi mākou iā \( 3x \) ma ke ʻano he loli hou \( u \), kahi \( u = 3x \). A laila ua like ʻo \( x \to 0 \) me \( u \to 0 \):
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{u \to 0} \frac{\sin(u)}{u/3} = 3 \lim_{u \to 0} \frac{\sin(u)}{u} = 3 \cdot 1 = 3
\]
No laila:
\[
lim_{x 0} \frac{\sin(3x)}{x} = 3
\]
Ka hopena
ʻO ka palena o kahi hana he manaʻo nui ia i ka calculus e kōkua iā mākou e hoʻomaopopo i ke ʻano o kahi hana ma kahi kikoʻī. Ma o kēia mau laʻana a me nā kūkākūkā ʻana, ua hoʻopili mākou i nā ʻano like ʻole o nā palena, e like me ka hoʻohui, ka hoʻonui ʻana, a me ka mahele ʻana, a me ka hoʻopili ʻana o ke kānāwai o L'Hôpital a me ka hoʻololi loli. He mea nui ka hoʻomaopopo ʻana i kēia manaʻo no nā haʻawina calculus holomua a me kāna mau noi ma nā ʻano ʻepekema a me ka ʻenekinia like ʻole.
ʻO ka hoʻomaopopo ʻana i nā waiwai o nā palena hana e hiki ai iā mākou ke kālailai a hoʻoponopono i nā ʻano pilikia makemakika like ʻole me ka ʻoi aku ka maikaʻi a me ka pono. Me ka hoʻomaʻamaʻa mau ʻana, e lilo ka hoʻomaopopo ʻana i kēia mau manaʻo i mea maʻalahi a maʻalahi hoʻi e hoʻopili.
