Ajụjụ na Mkparịta ụka gbasara Njirimara nke Oke Ọrụ
Pendahuluan
Oke ọrụ bụ echiche dị mkpa na mgbakọ na mwepụ nke na-arụ ọrụ dị oke mkpa na nyocha mgbakọ na mwepụ na ọtụtụ ojiji sayensị. Oke ọrụ na-enyere anyị aka ịghọta omume nke ọrụ ka mgbanwe na-eru nso uru ụfọdụ. Ọtụtụ njirimara nke oke ọrụ na-enye ngwaọrụ maka ịgbakọ na ijikwa oke ngwa ngwa. N'isiokwu a, anyị ga-atụle ọtụtụ nsogbu atụ ma tụlee njirimara nke oke ọrụ.
Njirimara nke Oke Ọrụ
Tupu anyị abanye n'ime nsogbu ihe atụ, ka anyị lelee ụfọdụ njirimara bụ isi nke oke ọrụ ndị a na-ejikarị eme ihe:
1. Oke nke Mgbakwunye
\[
\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
\]
2. Oke mmụba
\[
\lim_{x \to a}[f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)
\]
3. Oke nkesa
\[
\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{nyere } \lim_{x \to a} g(x) \neq 0
\]
4. Oke nha na-adịgide adịgide
\[
\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)
\]
5. Oke njirimara
\[
\lim_{x \na a} x = a
\]
6. Oke nke Ọrụ Na-adịgide Adịgide
\[
\lim_{x \to a} c = c, \quad \text{ebe c bụ ihe na-agbanwe agbanwe}
\]
Site n'ịghọta ihe ndị a bụ isi, ka anyị tinye ha n'ọrụ na ụfọdụ nsogbu ihe atụ.
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ajụjụ Ihe Nlereanya nke 1
Nye ihe si na ya pụta:
\[
\lim_{x \to 3} (2x^2 + 5x – 1)
\]
Azịza:
Iji dozie oke a, anyị nwere ike itinye uru x = 3 ozugbo na ọrụ ahụ n'ihi na ọrụ a bụ polynomial na polynomials na-aga n'ihu n'ofe ngalaba ha.
\[
\lim_{x \to 3} (2x^2 + 5x – 1) = 2(3)^2 + 5(3) – 1
\]
Gụọ otu nzọụkwụ site na nzọụkwụ:
\[
= 2(9) + 15 – 1 = 18 + 15 – 1 = 32
\]
Ya mere:
\[
\lim_{x \to 3} (2x^2 + 5x – 1) = 32
\]
Ajụjụ Ihe Nlereanya nke 2
Ọnụọgụ:
\[
\lim_{x \to -2} \frac{3x^3 + 4x + 2}{x + 2}
\]
Azịza:
N'ihe atụ a, itinye x = -2 ozugbo n'ụdị nkebi ga-emepụta ụdị a na-anaghị ekpebi \( \frac{0}{0} \), yabụ anyị kwesịrị ịgbakọ ya n'ụzọ ọzọ. Otu ụzọ bụ site n'ịkọwa ọnụọgụgụ.
Kọwaa ihe ngụkọ \( 3x^3 + 4x + 2 \):
Site n'ịnwale uru nke \( x = -2 \) na nkewa fọdụrụ, anyị ga-enweta:
\[
3(-2)^3 + 4(-2) + 2 = -24 – 8 + 2 = -30 \quad \text{(yabụ, enweghị ike ịtụle nke a nke ọma na-enweghị enyemaka nke ụzọ ndị ọzọ)}
\]
Nke a na-egosi na usoro nhazi kpọmkwem nwere ike ọ gaghị arụ ọrụ nke ọma. Ma ọ bụghị ya, anyị nwere ike ịnwa usoro L'Hôpital. Ọ bụrụ na anyị ekewaa ọnụọgụgụ na nkewa:
Onye na-egosi ọnụọgụgụ: \( 3x^3 + 4x + 2 \) na-eme ka ọ dị iche na \( 9x^2 + 4 \).
Ihe nkesa: \( x + 2 \) na-ekewapụ na \( 1 \).
Mgbe ahụ tinye L'Hôpital:
\[
\lim_{x \to -2} \frac{9x^2 + 4}{1} = 9(-2)^2 + 4 = 9(4) + 4 = 36 + 4 = 40
\]
Ya mere:
\[
\lim_{x \to -2} \frac{3x^3 + 4x + 2}{x + 2} = 40
\]
Ajụjụ Ihe Nlereanya nke 3
Chọta:
\[
\lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4}
\]
Azịza:
Maka nsogbu oke mgbe \( x \to \infty \), anyị nwere ike kewaa ihe ọ bụla site na ogo kachasị elu nke x na denominator, nke bụ \( x^2 \).
\[
\lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4} = \lim_{x \to \infty} \frac{5 – \frac{2}{x} + \frac{3}{x^2}}{1 + \frac{4}{x^2}}
\]
Maka na mgbe \( x \to \infty \), \( \frac{1}{x} \to 0 \) na \( \frac{1}{x^2} \to 0 \), mgbe ahụ:
\[
\lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4} = \frac{5 – 0 + 0}{1 + 0} = 5
\]
Yabụ,
\[
\lim_{x \to \infty} \frac{5x^2 – 2x + 3}{x^2 + 4} = 5
\]
Ajụjụ Ihe Nlereanya nke 4
Nye ihe si na ya pụta:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x}
\]
Azịza:
Anyị maara site na njirimara nke oke na:
\[
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]
Ugbua, anyị na-eji \(3x\) dochie ya dị ka mgbanwe ọhụrụ \(u\), ebe \(u = 3x\). Mgbe ahụ \(x \to 0 \) bụ ihe nhata na \(u \to 0 \):
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{u \to 0} \frac{\sin(u)}{u/3} = 3 \lim_{u \to 0} \frac{\sin(u)}{u} = 3 \cdot 1 = 3
\]
Ya mere:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = 3
\]
Mmechi
Oke ọrụ bụ echiche dị mkpa na mgbakọ na mwepụ nke na-enyere anyị aka ịghọta omume nke ọrụ n'otu ebe. Site na ihe atụ na mkparịta ụka ndị a, anyị etinyela ọtụtụ ihe onwunwe nke oke, dịka mgbakwunye, mmụba, na nkewa, yana itinye iwu L'Hôpital na mgbanwe mgbanwe. Ịghọta echiche a dị oke mkpa maka ọmụmụ ihe mgbakọ na mwepụ dị elu na ojiji ya na ngalaba sayensị na injinia dị iche iche.
Ịmụta njirimara nke oke ọrụ na-enye anyị ohere inyocha ma dozie ọtụtụ nsogbu mgbakọ na mwepụ nke ọma na nke ọma. Site na omume mgbe niile, nghọta echiche ndị a ga-aghọ ihe a na-aghọtakwu ma dị mfe itinye n'ọrụ.
