Ngā Tauira Pātai e Matapaki ana i te Whakamahinga o ngā Herenga Mahi
Ko te rohenga o tētahi mahi he ariā taketake i roto i te tātaitai, e whakamahia ana hei whakatau i te whanonga o tētahi mahi i a ia e whakatata atu ana ki tētahi pūwāhi motuhake. I roto i te pāngarau, inā koa ko te tātaitai, he mea nui te mārama ki te rohenga o tētahi mahi hei whakatū i te turanga mō ētahi atu ariā pēnei i ngā taupatupatu me ngā taunga whakauru. Ka hipokina e tēnei tuhinga ngā tauira rapanga me te matapaki i ngā tono o ngā mahi rohenga hei whakarato i tētahi māramatanga hohonu ake mō tēnei kaupapa.
Whakataki ki ngā Herenga Mahi
Ko te rohe o tētahi mahi e whakaahua ana i te uara e whakatata atu ana te mahi i te whakatata atu o te taurangi ki tētahi uara. E rua ngā momo rohe e kōrerohia pinepinetia ana: ko ngā rohe taha-kotahi (te rohe maui me te rohe matau) me ngā rohe taha-rua. Ko te tuhi whānui mō te rohe o tētahi mahi \( f(x) \) i te whakatata atu o \( x \) \( a \) ko:
\[
\lim_{x \to a} f(x)
\]
Tauira Pātai 1: Te Herenga Taketake
Pātai:
Whakatauhia te uara o \(\lim_{x \to 2} (3x + 1)\).
Kōrero:
He tauira tēnei o tētahi rohe taketake ina he mahi rārangi te mahi \( f(x) = 3x + 1 \) e haere tonu ana puta noa i tōna rohe. Kātahi ka taea e tātou te whakakapi tika i te uara o \( x = 2 \) ki roto i te mahi.
\[
\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 6 + 1 = 7
\]
Nō reira, \(\lim_{x \to 2} (3x + 1) = 7\).
Tauira Pātai 2: Te Whakawhāiti mā te Wehenga mā te Kore
Pātai:
Whakatauhia te uara o \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3}\).
Kōrero:
Ki te whakakapi tika tātou i te \( x = 3 \) ki roto i te mahi, ka whiwhi tātou i te āhua kore-taurite \(\frac{0}{0}\). Nō reira, me whakangawari tātou i te mahi i te tuatahi.
Kia mōhio koe ko te taupū \( x^2 – 9 \) he āhua tapawhā ka taea te tauwehe:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
Nō reira, ka taea te tuhi anō i te mahi tuatahi penei:
\[
\frac{x^2 – 9}{x – 3} = \frac{(x – 3)(x + 3)}{x – 3}
\]
Mai i konei, ka taea e tātou te whakangawari mā te whakakore i te \( x – 3 \) i roto i te taupū me te tautō, mena ko \( x \neq 3 \):
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
Ka taea e tātou te tatau tika i te rohe mā te whakakapi i te \( x = 3 \):
\[
\lim_{x \to 3} (x + 3) = 3 + 3 = 6
\]
Nō reira, \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 6\).
Tauira 3: Ngā Rohe me ngā Mahi Hautau
Pātai:
Kimihia te uara o \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1}\).
Kōrero:
Ki te whakakapi tika tātou i te \( x = 1 \) ki roto i te mahi, ka whiwhi tātou i te āhua kore-taurite \(\frac{0}{0}\). Hei whakaoti i tēnei, me whakangawari tātou i te mahi. Ko tētahi huarahi ko te whakamārama i te taunga.
Ka whakareatia te taupū me te tauwehe ki te hononga o te taupū:
\[
\frac{\sqrt{x + 3} – 2}{x – 1} \cdot \frac{\sqrt{x + 3} + 2}{\sqrt{x + 3} + 2}
\]
Kātahi ka whiwhi tātou:
\[
\frac{(\sqrt{x + 3} – 2)(\sqrt{x + 3} + 2)}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{(x + 3) – 4}{(x – 1)(\sqrt{x + 3} + 2)}
\]
Whakangāwaritia te taunga:
\[
x + 3 – 4 = x – 1
\]
Nō reira:
\[
\frac{x – 1}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{1}{\sqrt{x + 3} + 2}
\]
Ka taea e tātou te tatau i te rohe mā te whakakapinga \( x = 1 \):
\[
\lim_{x \to 1} \frac{1}{\sqrt{x + 3} + 2} = \frac{1}{\sqrt{1 + 3} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Nō reira, \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} = \frac{1}{4}\).
Tauira Pātai 4: Ngā Herenga me te Pāngatoru
Pātai:
Whakatauhia te uara o \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).
Kōrero:
E mōhio ana tātou mō ngā rohe taketake o te ine whārite, koinei ngā rohe e mōhiotia whānuitia ana:
\[
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]
Mō tēnei raruraru, me hono tātou ki taua āhua taketake. Kia mahara ko \( 3x \) te tautohe o te sine. Ka taea e tātou te whakaatu i te rohenga mā te whakarerekē penei:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3
\]
Nā te mea ko \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \) me \( u = 3x \), nō reira:
\[
\lim_{x \to 0} \frac{\sin(3x)}{3x} = 1
\]
Nā reira:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = 1 \cdot 3 = 3
\]
Nō reira, \(\lim_{x \to 0} \frac{\sin(3x)}{x} = 3\).
Whakamutunga
Kua kapi i tēnei tuhinga ētahi tauira raruraru, ā, kua matapakihia te whakamahinga o ngā rohe mahi i roto i te tātaitai. I roto i ia tauira raruraru, ka tīmata te matapakinga mā te tautuhi i te āhua i whiwhihia i te wā e whakakapia ana ngā uara, kātahi ka tūhura i ngā huarahi hei whakangawari, hei whakamārama rānei i te mahi. He mea nui te mārama ki ngā rohe mahi me te pehea e whakaoti ai i aua rohe hei matatau ki ngā ariā pāngarau matatau, pērā i ngā taupatupatu me ngā tau whakauru. Mā te mahi tonu, ka kaha ake, ka hohonu ake tō mārama ki ngā rohe mahi.
