Jawaban:
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } = 1[/tex]
Penjelasan dengan langkah-langkah:
Limit fungsi tak hingga
sifat limit tak hingga bentuk pecahan
[tex]\boxed{\displaystyle\sf\lim _{x\to \infty }\frac{ax^m +bx^{m-1} + cx^{m-2} +\cdots}{px^n + qx^{n-1} + rx^{n-2}+\cdots}=}[/tex]
Kembali pada soal
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } = ...[/tex]
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } \: (a \: = 1 \: . \: p \: = \: 1) \: dan \: m = n[/tex]
sehingga hasil dari limit tersebut adalah
[tex] = \frac{1}{1} [/tex]
[tex] = 1[/tex]
============================
Menggunakan L'Hopital
[tex]\begin{aligned}\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } &= \sf\lim _{x\to \infty }\frac{ \frac{d}{dx} (x ^{2} - 3x - 10) }{ \frac{d}{dx} ({x}^{2} - 5) }\\&=\sf\lim _{x\to \infty }\frac{2x-3}{ 2x }\\&=\sf\lim _{x\to \infty }\frac{ \frac{d}{dx} (2x-3) }{ \frac{d}{dx} (2x) }\\&=\sf\lim _{x\to \infty }\frac{2}{2}\\&=\sf 1\end{aligned}[/tex]
==============================
Pelajari lebih lanjut materi limit fungsi lainnya :
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } = \bold{1}[/tex]
Metode L'Hôpital
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 }[/tex]
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{(1 \times 2 \times x ^{2 - 2}) - (0 \times 1 \times 3 {x}^{1 - 2}) - (0 \times 10 )}{ (1 \times 2 \times {x}^{2 - 2}) - (0 \times 5 )}[/tex]
[tex] = \displaystyle\frac{2x ^{0} - 0 - 0 }{ 2{x}^{0} - 0 }[/tex]
[tex] = \displaystyle \frac{2 }{ 2 }[/tex]
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Jawaban:
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } = 1[/tex]
Penjelasan dengan langkah-langkah:
Limit fungsi tak hingga
sifat limit tak hingga bentuk pecahan
[tex]\boxed{\displaystyle\sf\lim _{x\to \infty }\frac{ax^m +bx^{m-1} + cx^{m-2} +\cdots}{px^n + qx^{n-1} + rx^{n-2}+\cdots}=}[/tex]
Kembali pada soal
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } = ...[/tex]
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } \: (a \: = 1 \: . \: p \: = \: 1) \: dan \: m = n[/tex]
sehingga hasil dari limit tersebut adalah
[tex] = \frac{1}{1} [/tex]
[tex] = 1[/tex]
============================
Menggunakan L'Hopital
[tex]\begin{aligned}\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } &= \sf\lim _{x\to \infty }\frac{ \frac{d}{dx} (x ^{2} - 3x - 10) }{ \frac{d}{dx} ({x}^{2} - 5) }\\&=\sf\lim _{x\to \infty }\frac{2x-3}{ 2x }\\&=\sf\lim _{x\to \infty }\frac{ \frac{d}{dx} (2x-3) }{ \frac{d}{dx} (2x) }\\&=\sf\lim _{x\to \infty }\frac{2}{2}\\&=\sf 1\end{aligned}[/tex]
==============================
Pelajari lebih lanjut materi limit fungsi lainnya :
Jawaban:
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 } = \bold{1}[/tex]
Penjelasan dengan langkah-langkah:
Metode L'Hôpital
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{x ^{2} - 3x - 10 }{ {x}^{2} - 5 }[/tex]
[tex]\displaystyle\sf\lim _{x\to \infty }\frac{(1 \times 2 \times x ^{2 - 2}) - (0 \times 1 \times 3 {x}^{1 - 2}) - (0 \times 10 )}{ (1 \times 2 \times {x}^{2 - 2}) - (0 \times 5 )}[/tex]
[tex] = \displaystyle\frac{2x ^{0} - 0 - 0 }{ 2{x}^{0} - 0 }[/tex]
[tex] = \displaystyle \frac{2 }{ 2 }[/tex]
[tex] = 1[/tex]