Ngā Āhuatanga o ngā Herenga Mahi
I roto i te pāngarau, he ariā taketake ngā rohe e whakamahia whānuitia ana i roto i te tātari me te tātaitai. Mā te rohe o tētahi mahi ka āwhina i te mārama ki te whanonga o tētahi mahi ina whakatata atu ki tētahi uara. Ehara i te mea he āwhina noa iho tēnei māramatanga i roto i ngā rangahau ariā engari he whānui hoki ngā tono mahi i roto i te pūtaiao me te hangarau. I roto i tēnei tuhinga, ka matapakihia e mātou ngā āhuatanga rerekē o te rohe o tētahi mahi me tōna hiranga i roto i te horopaki whānui.
Te Whakamāramatanga o te Here
I roto i ngā kupu māmā noa iho, ko te rohe o tētahi mahi ko te uara e whakatata atu ana te mahi i te whakatata atu o te taurangi whakauru ki tētahi pūwāhi. Mēnā ka tuhia e tātou te rohe o te mahi \( f(x) \) i te whakatata atu o \( x \) ki \( a \) hei \( L \), ka tuhia e tātou:
\[ \lim_{x \to a} f(x) = L. \]
Ko te tikanga tēnei, ka tata atu a \( x \) ki a \( a \), ka tata atu hoki te uara o \( f(x) \) ki a \( L \).
Ngā Āhuatanga Taketake o ngā Here
Ko ngā āhuatanga taketake o ngā rohe e whakamahia whānuitia ana i roto i te pāngarau ēnei e whai ake nei:
1. Tepe Pūmau:
\[ \lim_{x \to a} c = c, \]
ko te \( c \) he pūmau. Arā, ko te rohe o tētahi pūmau ko te uara o te pūmau tonu.
2. Ngā Here Tuakiri:
\[ \lim_{x \to a} x = a. \]
Ko te rohe o te taurangi tuakiri \( x \) i te whakatata atu o \( x \) ki \( a \) ko \( a \).
3. Te Rohe o te Tāpiri:
Mena ko \( \lim_{x \to a} f(x) = L \) me \( \lim_{x \to a} g(x) = M \), kātahi
\[ \lim_{x \to a} [f(x) + g(x)] = L + M. \]
Arā, ko te rohenga o te tapeke o ngā mahi e rua ko te tapeke o ngā rohenga o aua mahi.
4. Tepe Whakaiti:
\[ \lim_{x \to a} [f(x) – g(x)] = L – M. \]
Ko te rohenga o te tangohanga o ngā mahi e rua ko te tangohanga o ngā rohenga o aua mahi.
5. Tepe Whakarea:
\[ \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M. \]
Ko te rohenga o te hua o ngā mahi e rua ko te hua o ngā rohenga o aua mahi.
6. Tepe Tohatoha:
Mena ko \( M \neq 0 \), kātahi
\[ \lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{L}{M}. \]
Ko te rohenga o te wehenga o ngā mahi e rua ko te wehenga o ngā rohenga o aua mahi.
7. Tepe Tūnga:
Mō ngā mana tauoti pai \( n \),
\[ \lim_{x \to a} [f(x)]^n = [ \lim_{x \to a} f(x) ]^n. \]
Ko te tikanga o tēnei ka taea e tātou te nuku i te mahi taupū i waho o te tukanga tango-here.
Ngā Āhuatanga Motuhake
Haunga ngā āhuatanga taketake o runga ake nei, he maha atu anō ngā āhuatanga motuhake e whakawhirinaki ana ki ētahi āhuatanga:
1. Tepe Mahi Whakakotahi:
Mena ko \( \lim_{x \to a} g(x) = b \) me \( \lim_{x \to b} f(y) = L \), ko \( y = g(x) \), kātahi
\[ \lim_{x \to a} f(g(x)) = L. \]
I tēnei wā, ka aromatawaihia e tātou te rohe o tētahi mahi whakahiato mā te aromatawai i te rohe o te mahi o roto i te tuatahi, kātahi ko te mahi o waho.
2. Te Motunga Mutunga Kore:
Ki te nui haere te uara o \( f(x) \) i te whakatata atu o \( x \) ki \( a \), ka tuhia e mātou:
\[ \lim_{x \to a} f(x) = \infty. \]
E tohu ana tēnei ka tipu haere te mahi me te kore here i a ia e whakatata atu ana ki tētahi pūwāhi.
Tepe Taha
Kāore e oti te kōrero mō ngā rohe me te kore e whakaarohia ngā rohe taha, arā, ngā rohe i te taha maui (te rohe maui) me te taha matau (te rohe matau):
1. Te rohe mai i te taha maui:
\[ \lim_{x \to a^-} f(x) = L, \]
ina whakatata atu a \( x \) ki a \( a \) mai i tētahi uara iti ake, ki maui rānei.
2. Te rohe mai i te taha matau:
\[ \lim_{x \to a^+} f(x) = L, \]
ina whakatata atu a \( x \) ki a \( a \) mai i tētahi uara nui ake, ki te taha matau rānei.
Ka noho te rohe \( \lim_{x \to a} f(x) \) mēnā he ōrite ngā rohe mai i te taha maui me te taha matau.
Ngā Whakamahinga o ngā Here i te Ao Tūturu
He ariā nui te herenga e whakamahia ana i roto i ngā āhuatanga maha o te ao tūturu, tae atu ki:
1. Ahupūngao:
I roto i te ahupūngao, ka whakamahia ngā rohe hei tautuhi i ngā ariā pēnei i te tere inamata me te whakaterenga. Hei tauira, ko te tere inamata te rohe o te tere toharite ina tata te wā ki te kore.
2. Ōhanga:
Ka whakamahia hoki ngā here i roto i te ōhanga hei tatau i te tere huringa taha, pērā i te painga taha, te utu taha rānei i te mea e tata ana te nui o te whakaputanga ki tētahi uara motuhake.
3. Tikanga:
I roto i te hangarau, ka whakamahia ngā rohe mō te tātari pumau, te whakahaere pūnaha, me te whakatauira i ngā pūnaha hihiri matatini.
Whakamutunga
He ariā nui ngā rohe i roto i te tātaitai me te tātari pāngarau e āwhina ana i a tātou ki te mārama ki te whanonga o ngā mahi ina whakatata atu ki ētahi uara. He tino whai hua ngā āhuatanga o ngā rohe mō te whakahaere i ngā tātaitai me te tātari atu i roto i ngā momo mara o te pūtaiao me te hangarau. Mā te mārama ki ngā momo āhuatanga o ngā rohe ka homai he taputapu kaha ki a tātou mō te tūhura me te whakatauira i ngā āhuatanga uaua o te ao tūturu.
