Pembuktian Sifat Limit Fungsi Trigonometri

Pembuktian Sifat-Sifat Limit Fungsi Trigonometri

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Edumatik.Net – Cukup banyak yang mencari pembuktian sifat-sifat limit fungsi trigonometri, jika kamu termasuk salah satunya maka kamu sudah berada di tempat yang benar, karena melalui tulisan ini aku akan jelasin sifat-sifat limit fungsi trigonometri dan pembuktiannya.

Pembuktian sifat-sifat limit fungsi trigonometri membutuhkan beberapa rumus dasar diantaranya rumus luas juring, rumus luas segitiga, trigonometri dasar, dan teorema apit.

Pembuktian sifat limit fungsi trigonometri memang sedikit membuat otak berasap, tapi bukan berarti tidak bisa dipelajari. Nah oleh karena itu, biar kamu paham aku akan jelasin step by step pembuktian sifat limit fungsi trigonometri ini.

Berikut ini adalah rumus atau teori-teori pendukung untuk membuktikan sifat-sifat limit fungsi trigonometri.

Teorema Apit Limit Fungsi Trigonometri

Misalkan \(f, g,\) dan \(h\) fungsi yang terdefinisikan pada interval terbuka \(I\) yang memuat \(a\) kecuali mungkin di \(a\) itu sendiri, sihingga \(f(x) \leq g(x) \leq h(x)\) untuk setiap \(x \in I\), \(x \neq a\).

Jika \(\displaystyle \lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L\), maka \(\displaystyle \lim_{x \to a} g(x) = L\).

atau penulisannya bisa seperti ini

\(\displaystyle \lim_{x \to a} f(x) \leq \lim_{x \to a} g(x) \leq \lim_{x \to a} h(x)\)

\(\displaystyle L \leq \lim_{x \to a} g(x) \leq L \)

Artinya nilai \(\displaystyle \lim_{x \to a} g(x) = L\)

Rumus Luas Segitiga

\(L = \displaystyle \frac{1}{2} . a . t\)

Rumus Luas Juring

\(\displaystyle Lj = \frac{\angle AOB . r^{2}}{2}\)

Pembuktian Sifat-Sifat Limit Fungsi Trigonometri

Perhatikan gambar berikut!

Perhatikan segitiga \(AOD\)

\(\displaystyle \begin{aligned} \sin x &= \frac{de}{mi} \\ &= \frac{AD}{OA} \end{aligned}\)
\(\displaystyle \sin x = \frac{AD}{r}\) . . . (1)

Perhatikan segitiga \(BOC\)

\(\displaystyle \begin{aligned} \tan x &= \frac{de}{sa} \\ &= \frac{BC}{OB} \end{aligned}\)
\(\displaystyle \tan x = \frac{BC}{r}\) . . . (2)

Sekarang lihat juring \(AOB\)

\(\displaystyle Lj = \frac{\angle AOB . r^{2}}{2}\)

\(\displaystyle Lj = \frac{x . r^{2}}{2}\)

Jika kita perhatikan dengan lebih teliti, luas segitiga \(AOB\) lebih kecil dari luas juring \(AOB\), sedangkan luas juring \(AOB\) lebih kecil dari luas segitiga \(BOC\). Sehingga bisa kita tuliskan sebagai berikut!

\(L \triangle AOB < \text{Lj AOB} < L \triangle BOC\)

\(\displaystyle \frac{1}{2} . OB. AD < \frac{xr^{2}}{2} < \frac{1}{2} .OB.BC\)

\(\displaystyle \frac{1}{2} . r. AD < \frac{xr^{2}}{2} < \frac{1}{2} . r .BC\)

Kalikan semuanya dengan \(\displaystyle \frac{2}{r^{2}}\)

\(\displaystyle \frac{AD}{r} < x < \frac{BC}{r}\)

\(\sin x < x < \tan x\) . . . (3)

1). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{x}{\sin x} = 1}\)

Rumus terakhir diatas atau pernyataan (3) dapat dikembangkan untuk membuktikan sifat-sifat limit fungsi trigonometri.

\(\sin x < x < \tan x\) . . . (bagi semuanya dengan \(\sin x\))

\(\displaystyle 1 < \frac{x}{\sin x} < \frac{\tan x}{\sin x}\)

Ingat rumus trigonometri dasar, \(\displaystyle \tan x = \frac{\sin x}{\cos x}\)

\(\displaystyle 1 < \frac{x}{\sin x} < \frac{\sin x}{\cos x} . \frac{1}{\sin x}\)

\(\displaystyle 1 < \frac{x}{\sin x} < \frac{1}{\cos x}\)

Jika semuanya menggunakan \(\displaystyle \lim_{x \to 0}\) maka akan menjadi seperti berikut.

\(\displaystyle \lim_{x \to 0} 1 < \lim_{x \to 0} \frac{x}{\sin x} < \lim_{x \to 0} \frac{1}{\cos x}\)

\(\displaystyle 1 < \lim_{x \to 0} \frac{x}{\sin x} < \frac{1}{\cos 0}\)

\(\displaystyle 1 < \lim_{x \to 0} \frac{x}{\sin x} < \frac{1}{1}\)

\(\displaystyle 1 < \lim_{x \to 0} \frac{x}{\sin x} < 1\)

Ingat “teorema limit apit” diatas!
Jadi kesimpulannya adalah sebagai berikut:

\(\displaystyle \lim_{x \to 0} \frac{x}{\sin x} = 1\) . . . (terbukti)

Oh ya lupa, agar kamu tidak terlalu pusing sebaiknya kamu pahami dulu materi limit fungsi aljabar. Nah sekarang kita kembangkan lagi rumus yang sudah dibuktikan barusan, kita akan mengubah-ngubah bentuknya tapi nilai atau maknanya tetap sama.

2). Sifat \(\displaystyle \color {red}{\lim_{x \to 0} \frac{\sin x}{x} = 1}\)

Dari sifat pertama, yaitu: \(\displaystyle \lim_{x \to 0} \frac{x}{\sin x} = 1\)

\(\displaystyle \lim_{x \to 0} \frac{x}{\sin x} = \lim_{x \to 0} 1\)

Pindah ruaskan keduanya, sehingga menjadi

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{x}{\sin x}}\)

Untuk memudahkan pemahaman kamu, dikarenakan bertanda “\(=\)” penulisannya boleh kita tukar.

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{x}{\sin x}} = \frac{1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{x}{\sin x}} = \displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{1}{\displaystyle \frac{x}{\sin x}} \right] = \lim_{x \to 0} \frac{1}{1}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{\sin x}{x} . 1 \right] = \lim_{x \to 0} 1\)

\(\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1\) . . . (terbukti)

3). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{x}{\tan x} =1}\)

Kembali ke pernyataan (3)

\(\sin x < x < \tan x\) (bagi semuanya dengan \(\tan x\))

\(\displaystyle \frac{\sin x}{\tan x} < \frac{x}{\tan x} < 1\)

Ingat, \(\displaystyle \tan x = \frac{\sin x}{\cos x}\)

\(\displaystyle \frac{\sin x}{\displaystyle \frac{\sin x}{\cos x}} < \frac{x}{\tan x} < 1\)

\(\displaystyle \sin x . \frac{\cos x}{\sin x} < \frac{x}{\tan x} < 1\)

\(\displaystyle \cos x < \frac{x}{\tan x} < 1\)

Tambahkan \(\displaystyle \lim_{x \to 0}\) kesemuanya.

\(\displaystyle \lim_{x \to 0} \cos x < \lim_{x \to 0} \frac{x}{\tan x} < \lim_{x \to 0} 1\)

\(\displaystyle 1 < \lim_{x \to 0} \frac{x}{\tan x} < 1\)

Berdasarkan teorema apit, kesimpulannya adalah sebagai berikut.

\(\displaystyle \lim_{x \to 0} \frac{x}{\tan x} =1\) . . . (terbukti)

Kita gunakan cara yang sama untuk menemukan bentuk yang terbalik.

4). Sifat \(\displaystyle \color {red}{\lim_{x \to 0} \frac{\tan x}{x} = 1}\)

Dari sifat ketiga, yaitu: \(\displaystyle \lim_{x \to 0} \frac{x}{\tan x} =1\)

\(\displaystyle \lim_{x \to 0} \frac{x}{\tan x} = \lim_{x \to 0} 1\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{x}{\tan x}}\)

atau

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{x}{\tan x}} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{x}{\tan x}} = \displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{1}{\displaystyle \frac{x}{\tan x}} \right] = \lim_{x \to 0} \frac{1}{1}\)

\(\displaystyle \lim_{x \to 0} \frac{\tan x}{x} . 1 = \lim_{x \to 0} 1\)

\(\displaystyle \lim_{x \to 0} \frac{\tan x}{x} = 1\) . . . (terbukti)

Empat sifat limit trigonometri sudah kita buktikan, sekarang kita buktikan yang kelima berdasarkan sifat-sifat yang sudah dibuktikan sebelumnya.

5). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{ax}{\sin ax} = 1}\)

Ingat \(\displaystyle \lim_{x \to 0} \frac{x}{\sin x} = 1\), artinya berlaku juga untuk bentuk \(\displaystyle \lim_{y \to 0} \frac{y}{\sin y} = 1\). Setujukan?

Misalkan \(y = ax\). Nilai \(y\) bergantung pada nilai \(x\), ketika \(x\) mendekati nilai \(0\) maka \(y\) akan mendekati nilai \(0\).

Makna lainnya bisa seperti ini, ketika \(x\) mendekati nilai \(0\) maka nilai \(ax\) akan mendekati \(0\) juga.

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{ax}{\sin ax} &= \lim_{x \to 0} \frac{ax}{\sin ax} \\ &= \lim_{ax \to a.0} \frac{ax}{\sin ax} \\ &= \lim_{ax \to 0} \frac{ax}{\sin ax} \\ &= \lim_{y \to 0} \frac{y}{\sin y} \end{aligned}\)

\(\displaystyle \lim_{x \to 0} \frac{ax}{\sin ax} = 1\) . . . (terbukti)

6). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\sin ax}{ax} = 1}\)

Dari sifat kelima, yaitu: \(\displaystyle \lim_{x \to 0} \frac{ax}{\sin ax} = 1\)

\(\displaystyle \lim_{x \to 0} \frac{ax}{\sin ax} = \lim_{x \to 0} 1\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{ax}{\sin ax}}\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{ax}{\sin ax}} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{ax}{\sin ax}} = \displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{1}{\displaystyle \frac{ax}{\sin ax}} \right] = \lim_{x \to 0} \frac{1}{1}\)

\(\displaystyle \lim_{x \to 0} \frac{\sin ax}{ax} = 1\) . . . (terbukti)

Dengan menggunakan cara yang sama, maka didapatkan juga sifat limit trigonometri tangen.

7). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{ax}{\tan ax} = 1}\)

\(\displaystyle \lim_{x \to 0} \frac{x}{\tan x} = 1\), artinya berlaku juga untuk bentuk \(\displaystyle \lim_{y \to 0} \frac{y}{\tan y} = 1\).

Misalkan \(y = ax\), ketika \(x\) mendekati nilai \(0\) maka \(y\) akan mendekati nilai \(0\).

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{ax}{\tan ax} &= \lim_{x \to 0} \frac{ax}{\tan ax} \\ &= \lim_{ax \to a.0} \frac{ax}{\tan ax} \\ &= \lim_{ax \to 0} \frac{ax}{\tan ax} \\ &= \lim_{y \to 0} \frac{y}{\tan y} \end{aligned}\)

\(\displaystyle \lim_{x \to 0} \frac{ax}{\tan ax} = 1\) . . . (terbukti)

8). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\tan ax}{ax} = 1}\)

Dari sifat ketujuh, yaitu: \(\displaystyle \lim_{x \to 0} \frac{ax}{\tan ax} = 1\)

\(\displaystyle \lim_{x \to 0} \frac{ax}{\tan ax} = \lim_{x \to 0} 1\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{ax}{\tan ax}}\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{ax}{\tan ax}} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{ax}{\tan ax}} = \displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} 1}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{1}{\displaystyle \frac{ax}{\tan ax}} \right] = \lim_{x \to 0} \frac{1}{1}\)

\(\displaystyle \lim_{x \to 0} \frac{\tan ax}{ax} = 1\) . . . (terbukti)

Sekarang kita cari lagi sifat limit trigonometri lainnya, masih sanggup? Oke aku lanjutin.

9). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{ax}{\sin bx} = \frac{a}{b}}\)

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{ax}{\sin bx} &= \lim_{x \to 0} \frac{ax}{\sin bx} \\ &= \lim_{x \to 0} \left[ \frac{ax}{\sin bx} . \color{red}{1} \right] \\ &= \lim_{x \to 0} \left[ \frac{ax}{\sin bx} . \color{red}{\frac{bx}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\color{red}{bx}}{\sin bx} . \frac{ax}{\color{red}{bx}} \right] \\ &= \lim_{x \to 0} \frac{\color{red}{bx}}{\sin bx} . \lim_{x \to 0} \frac{ax}{\color{red}{bx}} \\ &= 1 . \lim_{x \to 0} \frac{a}{\color{red}{b}} \end{aligned}\)

\(\displaystyle \lim_{x \to 0} \frac{ax}{\sin bx} = \frac{a}{b}\) . . . (terbukti)

10). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b}}\)

Dari sifat kesembilan dengan koefisiennya ditukar tapi tidak mengubah bentuk, yaitu: \(\displaystyle \lim_{x \to 0} \frac{bx}{\sin ax} = \frac{b}{a}\)

\(\displaystyle \lim_{x \to 0} \frac{bx}{\sin ax} = \lim_{x \to 0} \frac{b}{a}\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{bx}{\sin ax}} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{b}{a}}\)

\(\displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{bx}{\sin ax}} = \displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{b}{a}}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{1}{\displaystyle \frac{bx}{\sin ax}} \right] = \displaystyle \lim_{x \to 0} \frac{1}{\displaystyle \frac{b}{a}}\)

\(\displaystyle \lim_{x \to 0} \frac{\sin ax}{bx} = \lim_{x \to 0} \frac{a}{b}\)

\(\displaystyle \lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b}\) . . . (terbukti)

11). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{ax}{\tan bx} = \frac{a}{b}}\)

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{ax}{\tan bx} &= \lim_{x \to 0} \frac{ax}{\tan bx} \\ &= \lim_{x \to 0} \left[ \frac{ax}{\tan bx} . \color{red}{1} \right] \\ &= \lim_{x \to 0} \left[ \frac{ax}{\tan bx} . \color{red}{\frac{bx}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\color{red}{bx}}{\tan bx} . \frac{ax}{\color{red}{bx}} \right] \\ &= \lim_{x \to 0} \frac{\color{red}{bx}}{\tan bx} . \lim_{x \to 0} \frac{ax}{\color{red}{bx}} \\ &= 1 . \lim_{x \to 0} \frac{a}{\color{red}{b}} \end{aligned}\)

\(\displaystyle \lim_{x \to 0} \frac{ax}{\tan bx} = \frac{a}{b}\) . . . (terbukti)

12). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\tan ax}{bx} = \frac{a}{b}}\)

Dari sifat kesebelas dengan koefisiennya ditukar tapi tidak mengubah bentuk, yaitu: \(\displaystyle \lim_{x \to 0} \frac{bx}{\tan ax} = \frac{b}{a}\)

\(\displaystyle \lim_{x \to 0} \frac{bx}{\tan ax} = \lim_{x \to 0} \frac{b}{a}\)

\(\displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{bx}{\tan ax}} = \displaystyle \frac{1}{\displaystyle \lim_{x \to 0} \frac{b}{a}}\)

\(\displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{bx}{\tan ax}} = \displaystyle \frac{\displaystyle \lim_{x \to 0} 1}{\displaystyle \lim_{x \to 0} \frac{b}{a}}\)

\(\displaystyle \lim_{x \to 0} \left[ \frac{1}{\displaystyle \frac{bx}{\tan ax}} \right] = \displaystyle \lim_{x \to 0} \frac{1}{\displaystyle \frac{b}{a}}\)

\(\displaystyle \lim_{x \to 0} \frac{\tan ax}{bx} = \lim_{x \to 0} \frac{a}{b}\)

\(\displaystyle \lim_{x \to 0} \frac{\tan ax}{bx} = \frac{a}{b}\) . . . (terbukti)

13). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\sin ax}{\sin bx} = \frac{a}{b}}\)

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{\sin ax}{\sin bx} &= \lim_{x \to 0} \left[ \frac{\sin ax}{\sin bx} . \color{red}{1} . \color{red}{1} \right] \\ &= \lim_{x \to 0} \left[ \frac{\sin ax}{\sin bx} . \color{red}{\frac{ax}{ax}} . \color{red}{\frac{bx}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\sin ax}{\color{red}{ax}} . \frac{\color{red}{bx}}{\sin bx} . \color{red}{\frac{ax}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\sin ax}{ax}. \frac{bx}{\sin bx} . \frac{a}{b} \right] \\ &= \frac{a}{b} . \lim_{x \to 0} \frac{\sin ax}{ax} . \lim_{x \to 0} \frac{bx}{\sin bx} \\ &= \frac{a}{b} . 1 . 1 \end{aligned}\)
\(\displaystyle \lim_{x \to 0} \frac{\sin ax}{\sin bx} = \frac{a}{b}\) . . . (terbukti)

14). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\tan ax}{\tan bx} = \frac{a}{b}}\)

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{\tan ax}{\tan bx} &= \lim_{x \to 0} \left[ \frac{\tan ax}{\tan bx} . \color{red}{1} . \color{red}{1} \right] \\ &= \lim_{x \to 0} \left[ \frac{\tan ax}{\tan bx} . \color{red}{\frac{ax}{ax}} . \color{red}{\frac{bx}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\tan ax}{\color{red}{ax}} . \frac{\color{red}{bx}}{\tan bx} . \color{red}{\frac{ax}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\tan ax}{ax}. \frac{bx}{\tan bx} . \frac{a}{b} \right] \\ &= \frac{a}{b} . \lim_{x \to 0} \frac{\tan ax}{ax} . \lim_{x \to 0} \frac{bx}{\tan bx} \\ &= \frac{a}{b} . 1 . 1 \end{aligned}\)
\(\displaystyle \lim_{x \to 0} \frac{\tan ax}{\tan bx} = \frac{a}{b}\) . . . (terbukti)

15). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\sin ax}{\tan bx} = \frac{a}{b}}\)

Kita akan gunakan sifat-sifat yang sudah dibuktikan sebelumnya untuk membuktikan sifat limit fungsi trigonometri ke-15.

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{\sin ax}{\tan bx} &= \lim_{x \to 0} \left[ \frac{\sin ax}{\tan bx} . \color{red}{1} . \color{red}{1} \right] \\ &= \lim_{x \to 0} \left[ \frac{\sin ax}{\tan bx} . \color{red}{\frac{ax}{ax}} . \color{red}{\frac{bx}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\sin ax}{\color{red}{ax}} . \frac{\color{red}{bx}}{\tan bx} . \color{red}{\frac{ax}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\sin ax}{ax}. \frac{bx}{\tan bx} . \frac{a}{b} \right] \\ &= \frac{a}{b} . \lim_{x \to 0} \frac{\sin ax}{ax} . \lim_{x \to 0} \frac{bx}{\tan bx} \\ &= \frac{a}{b} . 1 . 1 \end{aligned}\)
\(\displaystyle \lim_{x \to 0} \frac{\sin ax}{\tan bx} = \frac{a}{b}\) . . . (terbukti)

16). Sifat \(\displaystyle \color{red}{\lim_{x \to 0} \frac{\tan ax}{\sin bx} = \frac{a}{b}}\)

\(\displaystyle \begin{aligned} \lim_{x \to 0} \frac{\tan ax}{\sin bx} &= \lim_{x \to 0} \left[ \frac{\tan ax}{\sin bx} . \color{red}{1} . \color{red}{1} \right] \\ &= \lim_{x \to 0} \left[ \frac{\tan ax}{\sin bx} . \color{red}{\frac{ax}{ax}} . \color{red}{\frac{bx}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\tan ax}{\color{red}{ax}} . \frac{\color{red}{bx}}{\sin bx} . \color{red}{\frac{ax}{bx}} \right] \\ &= \lim_{x \to 0} \left[ \frac{\tan ax}{ax}. \frac{bx}{\sin bx} . \frac{a}{b} \right] \\ &= \frac{a}{b} . \lim_{x \to 0} \frac{\tan ax}{ax} . \lim_{x \to 0} \frac{bx}{\sin bx} \\ &= \frac{a}{b} . 1 . 1 \end{aligned}\)
\(\displaystyle \lim_{x \to 0} \frac{\tan ax}{\sin bx} = \frac{a}{b}\) . . . (terbukti)

Itulah pembahasan lengkap mengenai sifat limit fungsi trigonometri dan pembuktiannya. Besar harapan aku, kamu bisa paham artikel pembuktian sifat-sifat limit fungsi trigonometri ini. Berikutnya aku akan bahas sifat limit trigonometri dan contoh soalnya, bagikan tulisan ini agar bermanfaat untuk orang lain.

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