Pertemuan 4
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1) The document discusses the concept of limits in mathematics. A limit describes the value a function approaches as the input value gets closer to a certain point, without reaching it. 2) A limit is needed because functions are often not defined at certain points, but their values can still be determined by approaching those points. 3) To find the limit of a function f(x) as x approaches a value c, we determine if the left-hand limit and right-hand limit both exist and are equal. If so, that common value is the limit. If not, the limit does not exist.
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1) The document discusses the continuity of functions at a point and over an interval.
2) A function f is defined to be continuous at a point c if the limit of f(x) as x approaches c exists and is equal to f(c).
3) For a function to be continuous over an interval (a,b), it must be continuous at every point c within the interval.
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1) The document discusses limit theorems and trigonometric function limits. It introduces basic limit theorems including limits of constants, sums, products, quotients, and composite functions.
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Pertemuan 4
This document discusses the concept of limits in mathematics. It defines a limit as the value a function approaches as the input values get closer to a certain point, without actually reaching it. For a limit of a function f(x) as x approaches c to exist, the left-hand limit and right-hand limit must both exist and be equal to the same number L. A limit may not exist if the left and right sides are different, or if the values oscillate or approach infinity as x approaches c. The document also formally defines a limit using epsilon-delta notation.
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This document discusses limits and how to calculate them. It defines a limit as a number a function approaches as the input value approaches a certain number. It provides examples of using graphs and tables on a calculator to find limits, and discusses direct substitution and the replacement theorem for evaluating limits. Special cases like one-sided limits, limits of polynomials, and limits involving radicals are also covered.
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The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
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The document discusses limits, continuity, and related concepts. Some key points:
1) It defines the concept of a limit and explains how to evaluate one-sided and two-sided limits. A limit exists only if the left and right-sided limits are equal.
2) Continuity is defined as a function being defined at a point, and the limit existing and being equal to the function value. Functions like tan(x) are only continuous where the denominator is not 0.
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Pertemuan 4
This document discusses the concept of limits in mathematics. It defines a limit as the value a function approaches as the input values get closer to a certain point, without actually reaching it. For a limit of a function f(x) as x approaches c to exist, the left-hand limit and right-hand limit must both exist and be equal to the same number L. A limit may not exist if the left and right sides are different, or if the values oscillate or approach infinity as x approaches c. The document also formally defines a limit using epsilon-delta notation.
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This document discusses limits and how to calculate them. It defines a limit as a number a function approaches as the input value approaches a certain number. It provides examples of using graphs and tables on a calculator to find limits, and discusses direct substitution and the replacement theorem for evaluating limits. Special cases like one-sided limits, limits of polynomials, and limits involving radicals are also covered.
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The document discusses limits, continuity, and related concepts. Some key points:
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VISIT US @
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Pertemuan 4
- 1. MATEMATIKA DASAR Aswar Amiruddin, ST., MT.
- 3. Sub CP-MK Memahami konsep limit di suatu titik Indikator Menetukan nilai limit fungsi di satu titik dan sifatnya
- 4. Konsep Limit • Limit dapat diartikan sebagai menuju suatu batas, sesuatu yang dekat namun tidak dapat dicapai. • Mengapa harus ada limit? limit menjelaskan suatu fungsi jika batas tertentu didekati. • Mengapa harus didekati? karena suatu fungsi biasanya tidak terdefinisi pada titik- titik tertentu. • Walaupun suatu fungsi seringkali tidak terdefinisi untuk titik tertentu, namun masih dapat dicari tahu berapa nilai yang didekati oleh fungsi tersebut apabila titik tertentu semakin didekati yaitu dengan limit
- 5. Konsep Limit Misalkan f(x) suatu fungsi, c dan L bilangan real sedemikian hingga: • Bila x dekat c tetapi tidak sama dgn c (x c), maka f(x) dekat ke L • Bila x mendekati c tetapi x c, maka f(x) mendekati L • Misalkan f(x) dapat kita buat sedekat mungkin ke L dg membuat x cukup dekat c tetapi tidak sama dengan c • Maka dapat dikatakan bahwa limit f(x) bila x mendekati c adalah L, 𝐥𝐢𝐦 𝒙→𝒄 𝒇 (𝒙) = 𝑳
- 6. Konsep Limit Perhatikan fungsi berikut : 𝑓(𝑥) = 𝒙𝟐−𝟏 𝒙−𝟏 , terdefinisi untuk setiap x ϵ R kecuali untuk x = 1, fungsi tidak terdefinisi jika x=1, akan tetapi dengan mengambil nilai mendekati 1 maka fungsi di atas terdefinisi.
- 7. Konsep Limit Dari tabel di atas dan grafik disamping terlihat bahwa f(x) mendekati 2 jika x mendekati 1 Secara matematis dapat dituliskan Sebagai berikut 𝐥𝐢𝐦 𝒙→𝟏 𝒙𝟐 − 𝟏 𝒙 − 𝟏 = 𝟐
- 8. Pengertian limit secara intuitif Kita tuliskan lim 𝑥→𝑐 𝑓(𝑥) = 𝐿 apabila jika x dekat ke c, maka nilai f(x) dekat ke L lim 𝑥→𝑐 𝑓(𝑥) disebut “limit f di c”. Bilangan L merupakan nilai limit tersebut. Contoh 𝒍𝒊𝒎 𝒙→𝟏 𝒙𝟐 − 𝟏 𝒙 − 𝟏 = 𝟐
- 9. Cara mencari limit suatu fungsi 1. Disubtitusi langsung 2. Pemfaktoran 3. Kali sekawan (merasionalkan bentuk akar) lim 𝑥→1 3𝑥 + 5 = lim 𝑥→3 𝑥2 − 5𝑥 + 6 𝑥 − 3 = lim 𝑥→1 𝑥2 − 1 𝑥 − 1 = Contoh : lim 𝑥→2 2𝑥2 − 3𝑥 − 2 𝑥 − 2 =
- 10. Arti ”mendekati” pada konsep limit, bisa dari kiri titik c atau dari kanan titik c. Dengan kata lain, nilai limit f (x) untuk x mendekati c ada, jika limit kiri dan limit kanan ada. Limit Kanan dan Limit Kiri Limit kanan 𝑙𝑖𝑚 𝑥→𝑐+ 𝑓(𝑥) = 𝐿 berarti jika x > c dan dekat ke c, maka f(x) dekat ke L. Limit kanan 𝑙𝑖𝑚 𝑥→𝑐− 𝑓(𝑥) = 𝐿 berarti jika x < c dan dekat ke c, maka f(x) dekat ke L. Limit Kanan dan Limit Kiri
- 11. Contoh :
- 13. Bilakah Limit Ada/Tidak Ada? Limit fungsi f di c ada jika dan hanya jika limit kanan dan limit kiri f di c ada dan nilainya sama dengan L (perhatikan definisi limit). Limit f di c tidak ada bila salah satu di antara beberapa kemungkinan berikut terjadi: 1. Limit kanan dan limit kiri f di c ada, tetapi nilainya tidak sama. 2. Limit kanan atau limit kiri f di c tidak ada, karena a. Nilai f di dekat c menuju tak terhingga. b. Nilai f di dekat c berosilasi.
- 14. 𝒇 𝒙 = ቊ 𝟑 + 𝒙, 𝒙 < 𝟏 𝟑 − 𝒙, 𝒙 ≥ 𝟏 Contoh :