Ajụjụ Ihe Nlereanya Banyere Iji Ọrụ Eme Ihe n'Oke Ọrụ
Oke ọrụ bụ echiche dị mkpa na mgbakọ na mwepụ, nke a na-ejikarị ekpebi omume nke ọrụ ka ọ na-eru nso ebe a kapịrị ọnụ. Na mgbakọ na mwepụ, ọkachasị mgbakọ na mwepụ, ịghọta oke ọrụ dị oke mkpa maka ịtọ ntọala maka echiche ndị ọzọ dị ka ihe ndị sitere na ya na ihe ndị mejupụtara ya. Isiokwu a ga-ekpuchi nsogbu ihe atụ ma tụlee ojiji nke oke ọrụ iji nye nghọta miri emi nke isiokwu a.
Okwu Mmalite nke Oke Ọrụ
Oke ọrụ na-akọwa uru ọrụ na-abịaru nso ka mgbanwe ahụ na-erute uru ụfọdụ. E nwere ụdị oke abụọ a na-ekwukarị: oke otu akụkụ (oke aka ekpe na oke aka nri) na oke akụkụ abụọ. Ihe e dere n'ozuzu maka oke ọrụ \( f(x) \) dịka \( x \) si abịa \( a \) bụ:
\[
\lim_{x \to a} f(x)
\]
Ihe atụ Ajụjụ nke 1: Oke Nkịtị
Ajụjụ:
Chọpụta uru nke \(\lim_{x \to 2} (3x + 1)\).
Azịza:
Nke a bụ ihe atụ nke oke dị mkpa ebe ọrụ \( f(x) = 3x + 1 \) bụ ọrụ kwụ ọtọ nke na-aga n'ihu n'ofe ngalaba ya. Mgbe ahụ, anyị nwere ike dochie uru nke \( x = 2 \) ozugbo na ọrụ ahụ.
\[
\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 6 + 1 = 7
\]
Ya mere, \(\lim_{x \to 2} (3x + 1) = 7\).
Ihe atụ Ajụjụ nke 2: Oke site na Nkewa site na Zero
Ajụjụ:
Chọpụta uru nke \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3}\).
Azịza:
Ọ bụrụ na anyị etinye \( x = 3 \) ozugbo n'ime ọrụ ahụ, anyị ga-enweta ụdị nke a na-anaghị ekpebi \(\frac{0}{0}\). Ya mere, anyị ga-ebu ụzọ mee ka ọrụ ahụ dị mfe.
Rịba ama na ọnụọgụgụ \( x^2 – 9 \) bụ ụdị quadratic nke enwere ike ijikọta:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
Ya mere, enwere ike idegharị ọrụ mbụ ahụ dị ka:
\[
\frac{x^2 – 9}{x – 3} = \frac{(x – 3)(x + 3)}{x – 3}
\]
Site n'ebe a, anyị nwere ike ime ka ọ dị mfe site n'ịkagbu \( x – 3 \) na numerator na denominator, ma ọ bụrụhaala na \( x \neq 3 \):
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
Ugbu a, anyị nwere ike ịgbakọ oke ahụ ozugbo site n'itinye \( x = 3 \):
\[
\lim_{x \to 3} (x + 3) = 3 + 3 = 6
\]
Ya mere, \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 6\).
Ihe atụ nke 3: Oke nwere ọrụ nkebi
Ajụjụ:
Chọta uru nke \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1}\).
Azịza:
Ọ bụrụ na anyị etinye \( x = 1 \) ozugbo n'ime ọrụ ahụ, anyị ga-enweta ụdị a na-anaghị ekpebi \(\frac{0}{0}\). Iji dozie nke a, anyị kwesịrị ime ka ọrụ ahụ dị mfe. Otu ụzọ bụ ime ka ọnụọgụgụ ahụ dị mfe.
Anyị na-amụba ọnụọgụgụ na denominator site na iji njikọ nke ọnụọgụgụ:
\[
\frac{\sqrt{x + 3} – 2}{x – 1} \cdot \frac{\sqrt{x + 3} + 2}{\sqrt{x + 3} + 2}
\]
Mgbe ahụ anyị ga-enweta:
\[
\frac{(\sqrt{x + 3} – 2)(\sqrt{x + 3} + 2)}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{(x + 3) – 4}{(x – 1)(\sqrt{x + 3} + 2)}
\]
Mee ka ọnụọgụgụ dị mfe:
\[
x + 3 – 4 = x – 1
\]
Ka ọ were:
\[
\frac{x – 1}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{1}{\sqrt{x + 3} + 2}
\]
Ugbua, anyị nwere ike ịgbakọ oke ahụ site na nnọchi \( x = 1 \):
\[
\lim_{x \to 1} \frac{1}{\sqrt{x + 3} + 2} = \frac{1}{\sqrt{1 + 3} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Ya mere, \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} = \frac{1}{4}\).
Ajụjụ Ihe Nlereanya nke 4: Oke na Trigonometry
Ajụjụ:
Chọpụta uru nke \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).
Azịza:
Anyị maara na maka oke isi nke trigonometry, enwere oke ndị a a maara nke ọma:
\[
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]
Maka nsogbu a, anyị kwesịrị ijikọta ya na ụdị ahụ dị mkpa. Rịba ama na \(3x \) bụ arụmụka nke sine. Anyị nwere ike igosipụta oke site n'ịgbanwe ya dị ka ndị a:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3
\]
Maka na \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \) na \(u = 3x \), yabụ:
\[
\lim_{x \to 0} \frac{\sin(3x)}{3x} = 1
\]
Ya mere:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = 1 \cdot 3 = 3
\]
Ya mere, \(\lim_{x \to 0} \frac{\sin(3x)}{x} = 3\).
Mmechi
Edemede a ekpuchila ọtụtụ nsogbu ihe atụ ma tụlee ojiji nke oke ọrụ na mgbakọ na mwepụ. Na nsogbu ihe atụ ọ bụla, mkparịta ụka ahụ na-amalite site n'ịchọpụta ụdị e nwetara mgbe a na-agbanwe ụkpụrụ wee chọpụta ụzọ isi mee ka ọrụ ahụ dị mfe ma ọ bụ mee ka ọ dị mfe. Ịghọta oke ọrụ na otu esi edozi ha dị oke mkpa maka ịmụta echiche mgbakọ na mwepụ dị elu, dị ka ihe ndị sitere na ya na ihe ndị mejupụtara ya. Site na omume na-aga n'ihu, nghọta gị gbasara oke ọrụ ga-esiwanye ike ma dị omimi.
