Nā nīnau hoʻohālike e kūkākūkā ana i ka hoʻopili ʻana o nā palena hana
ʻO ka palena o kahi hana he manaʻo nui ia i ka calculus, i hoʻohana pinepine ʻia e hoʻoholo i ke ʻano o kahi hana i kona hoʻokokoke ʻana i kahi kiko kikoʻī. I ka makemakika, ʻoiai ka calculus, he mea nui ka hoʻomaopopo ʻana i ka palena o kahi hana no ka hoʻokumu ʻana i ke kahua no nā manaʻo hou aʻe e like me nā derivatives a me nā integrals. E uhi kēia ʻatikala i nā pilikia hoʻohālike a kūkākūkā i nā noi o nā hana palena e hāʻawi i kahi ʻike hohonu o kēia kumuhana.
Hoʻolauna i nā Palena Hana
ʻO ka palena o kahi hana e wehewehe ana i ka waiwai a ka hana e hoʻokokoke aku ai i ka wā e hoʻokokoke aku ai ke loli i kahi waiwai. Aia ʻelua ʻano palena i kūkākūkā pinepine ʻia: nā palena ʻaoʻao hoʻokahi (ka palena hema a me ka palena ʻākau) a me nā palena ʻaoʻao ʻelua. ʻO ka hōʻailona laulā no ka palena o kahi hana \( f(x) \) i ka wā e hoʻokokoke aku ai ʻo \( x \) \( a \) penei:
\[
lim_{x `to a} f(x)
\]
Laʻana Nīnau 1: Palena Kumu
Nīnau:
E hoʻoholo i ka waiwai o \(\lim_{x \to 2} (3x + 1)\).
Kūkākūkā:
He laʻana kēia o kahi palena kumu kahi i lilo ai ka hana \( f(x) = 3x + 1 \) i hana linear e hoʻomau ana ma kona kikowaena. A laila hiki iā mākou ke pani pololei i ka waiwai o \( x = 2 \) i loko o ka hana.
\[
lim_{x 2} (3x + 1) = 3(2) + 1 = 6 + 1 = 7
\]
No laila, \(\lim_{x \to 2} (3x + 1) = 7\).
Laʻana Nīnau 2: Palena ma ka Māhele ʻana ma ka ʻOle
Nīnau:
E hoʻoholo i ka waiwai o \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3}\).
Kūkākūkā:
Inā mākou e pani pololei iā \( x = 3 \) i loko o ka hana, e loaʻa iā mākou ke ʻano indeterminate \(\frac{0}{0}\). No laila, pono mākou e hoʻomaʻalahi mua i ka hana.
E hoʻomaopopo he ʻano quadratic ka numerator \( x^2 – 9 \) i hiki ke hoʻohālikelike ʻia:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
No laila, hiki ke kākau hou ʻia ka hana mua penei:
\[
\frac{x^2 – 9}{x – 3} = \frac{(x – 3)(x + 3)}{x – 3}
\]
Mai ʻaneʻi, hiki iā kākou ke hoʻomaʻalahi ma ka hoʻopau ʻana i ka \( x – 3 \) i loko o ka numerator a me ka denominator, inā ʻo \( x \neq 3 \):
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
I kēia manawa hiki iā mākou ke helu pololei i ka palena ma ke pani ʻana iā \( x = 3 \):
\[
\lim_{x \to 3} (x + 3) = 3 + 3 = 6
\]
No laila, \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 6\).
Laʻana 3: Nā Palena me nā Hana Hapa
Nīnau:
E huli i ka waiwai o \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1}\).
Kūkākūkā:
Inā mākou e pani pololei iā \( x = 1 \) i loko o ka hana, e loaʻa iā mākou ke ʻano indeterminate \(\frac{0}{0}\). No ka hoʻoponopono ʻana i kēia, pono mākou e hoʻomaʻalahi i ka hana. ʻO kekahi ala, ʻo ia ke hoʻākāka i ka helu.
Hoʻonui mākou i ka helu a me ka denominator e ka hui o ka helu:
\[
\frac{\sqrt{x + 3} – 2}{x – 1} \cdot \frac{\sqrt{x + 3} + 2}{\sqrt{x + 3} + 2}
\]
A laila loaʻa iā mākou:
\[
ʻO ka huaʻōlelo "x + 3" he like ia me ka huaʻōlelo "x + 3" a i ʻole "x + 3" a i ʻole "x" he like ia me ka huaʻōlelo ...
\]
E hoʻomaʻalahi i ka helu:
\[
x + 3 – 4 = x – 1
\]
No laila:
\[
\frac{x – 1}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{1}{\sqrt{x + 3} + 2}
\]
I kēia manawa hiki iā mākou ke helu i ka palena ma o ka pani ʻana \( x = 1 \):
\[
\lim_{x \to 1} \frac{1}{\sqrt{x + 3} + 2} = \frac{1}{\sqrt{1 + 3} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
No laila, \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} = \frac{1}{4}\).
Laʻana Nīnau 4: Nā Palena me ka Trigonometry
Nīnau:
E hoʻoholo i ka waiwai o \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).
Kūkākūkā:
Ua ʻike mākou no nā palena kumu o ka trigonometry, aia nā palena i ʻike nui ʻia:
\[
lim_{x 0} \frac{\sin(x)}{x} = 1
\]
No kēia pilikia, pono mākou e hoʻopili iā ia i kēlā ʻano kumu. E hoʻomaopopo ʻo \( 3x \) ka hoʻopaʻapaʻa o ka sine. Hiki iā mākou ke hōʻike i ka palena ma ka hoʻopunipuni ʻana iā ia penei:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3
\]
No ka mea, \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \) me \( u = 3x \), no laila:
\[
lim_{x 0} \frac{\sin(3x)}{3x} = 1
\]
No laila:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = 1 \cdot 3 = 3
\]
No laila, \(\lim_{x \to 0} \frac{\sin(3x)}{x} = 3\).
Ka hopena
Ua uhi kēia ʻatikala i kekahi mau pilikia hoʻohālike a ua kūkākūkā i ka hoʻopili ʻana o nā palena hana i ka calculus. Ma kēlā me kēia pilikia hoʻohālike, hoʻomaka ke kūkākūkā ma ka ʻike ʻana i ke ʻano i loaʻa i ka wā e pani ai i nā waiwai a laila e ʻimi i nā ala e hoʻomaʻalahi ai a hoʻākāka paha i ka hana. ʻO ka hoʻomaopopo ʻana i nā palena hana a pehea e hoʻoponopono ai iā lākou he mea nui ia no ka hoʻopaʻa ʻana i nā manaʻo makemakika holomua, e like me nā derivatives a me nā integrals. Me ka hoʻomaʻamaʻa mau, e ikaika a hohonu kou ʻike ʻana i nā palena hana.
