Soal 1. $ \lim_{x \rightarrow 3} \frac {x^2-9}{ \sqrt {10+2x}-x-1}=...$
$\lim_{x \rightarrow 3} \frac {x^2-9}{ \sqrt {10+2x}-x-1} \\ \lim_{x \rightarrow 3} \frac {x^2-9}{ \sqrt {10+2x}-(x+1)} \\ \text {Kalikan akar sekawan} \\ \lim_{x \rightarrow 3} \frac {x^2-9}{ \sqrt {10+2x}-(x+1)}. \frac {\sqrt {10+2x}+(x+1)}{\sqrt {10+2x}+(x+1)} \\ \lim_{x \rightarrow 3} \frac {x^2-9}{ 9-x^2} = -1$
Soal 2. $\lim_{x \rightarrow \infty } \sqrt {25x^2-9x+6} -5x+1=...$
$\lim_{x \rightarrow \infty } \sqrt {25x^2-9x+6} -5x+1= \lim_{x \rightarrow \infty } \sqrt {25x^2-9x+6} - (5x-1) \\ \lim_{x \rightarrow \infty } \sqrt {25x^2-9x+6} - \sqrt {(5x-1)^2} \\ \lim_{x \rightarrow \infty } \sqrt {25x^2-9x+6} - \sqrt {25x^2-10x+1} $
Penyelesaian
$ \frac {b-q}{2 \sqrt a} \\ \frac {-9-(-10)}{2 \sqrt {25}} = \frac {1}{10}$
Soal 3. $\lim_{x \rightarrow \infty } \sqrt {4x^2-2x+1} +\sqrt {x^2+x+6} -3x +6 =...$
$\lim_{x \rightarrow \infty } \sqrt {4x^2-2x+1} +\sqrt {x^2+x+6} -3x +6 \\ \lim_{x \rightarrow \infty } \sqrt {4x^2-2x+1} +\sqrt {x^2+x+6} -2x -x +6 \\ \lim_{x \rightarrow \infty } \sqrt {4x^2-2x+1} +\sqrt {x^2+x+6} +6 \\ \lim_{x \rightarrow \infty } \sqrt {4x^2-2x+1} -\sqrt {4x^2} +\sqrt {x^2+x+6} -\sqrt {x^2} +6 \\ \frac {-2-0}{2 \sqrt 4 }+ \frac {1-0}{\sqrt 1} +6 =6,5$
Soal 4. $\lim_{x \rightarrow 1 } \frac {(x^2-1) \sin 2 (x-1)}{-2 \sin ^2 (x-1)}=...$
$\lim_{x \rightarrow 1 } \frac {(x^2-1) \sin 2 (x-1)}{-2 \sin ^2 (x-1)} \\ \lim_{x \rightarrow 1 } \frac {(x-1)(x+1) \sin 2 (x-1)}{-2 \sin (x-1) \sin (x-1)} \\ \lim_{x \rightarrow 1 } \frac {(x+1)2}{-2} =-2$
Soal 5. $\lim_{x \rightarrow 2 } \frac {x^3 -4x}{x-2} =...$
$\lim_{x \rightarrow 2 } \frac {x^3 -4x}{x-2} \\ \lim_{x \rightarrow 2 } \frac {x (x-2)(x+2)}{x-2} \\ \lim_{x \rightarrow 2 } x(x+2) =8$
Soal 6. $\lim_{x \rightarrow 0 } \frac {2x \sin 3x}{1- \cos 6x}=... $
$ \lim_{x \rightarrow 0 } \frac {2x \sin 3x}{1- \cos 6x} \\ \lim_{x \rightarrow 0 } \frac {2x \sin 3x}{1- (1-2 \sin ^2 3x)} \\ \lim_{x \rightarrow 0 } \frac {2x \sin 3x}{2 \sin 3x \sin 3x} =\frac {1}{3}$
Soal 7. $\lim_{x \rightarrow 0 } \frac {1- \cos 2x}{x \tan \frac {1}{2}x} =...$
$\lim_{x \rightarrow 0 } \frac {1- \cos 2x}{x \tan \frac {1}{2}x} \\ \lim_{x \rightarrow 0 } \frac {1- (1 -2 \sin^2x)}{x \tan \frac {1}{2}x} \\ \lim_{x \rightarrow 0 } \frac {2 \sin x \sin x}{x \tan \frac {1}{2}x} =4$
Soal 8. $ \lim_{x \rightarrow 0 } \frac {\cos 2x}{\cos x - \sin x}=...$
$\lim_{x \rightarrow 0 } \frac {\cos 2x}{\cos x - \sin x} \\ \lim_{x \rightarrow 0 } \frac {\cos ^2x - \sin ^2x}{\cos x - \sin x} \\ \lim_{x \rightarrow 0 } \frac {(\cos x - \sin x)(\cos x + \sin x)}{\cos x - \sin x} \\ \lim_{x \rightarrow 0 } (\cos x + \sin x) = 1 $
Soal 9. $\lim_{x \rightarrow 0 } \frac {5x \tan x}{1- \cos 6x } =...$
$\lim_{x \rightarrow 0 } \frac {5x \tan x}{1- \cos 6x } \\ \lim_{x \rightarrow 0 } \frac {5x \tan x}{1- (1- 2\sin ^2 3x)} \\ \lim_{x \rightarrow 0 } \frac {5x \tan x}{2\sin 3x \sin 3x } = \frac {5}{18}$
Soal 10. $\lim_{x \rightarrow 0 } \frac {\cos 4x -1}{\cos x -1 }=...$
$\lim_{x \rightarrow 0 } \frac {\cos 4x -1}{\cos x -1 } \\ \lim_{x \rightarrow 0 } \frac {\cos 4x - \cos 0}{\cos x - \cos 0} \\ \lim_{x \rightarrow 0 } \frac {-2 \sin 2x \sin 2x} {-2 \sin \frac {1}{2}x \sin \frac {1}{2}x } =8$
Soal 11. $ \lim_{x \rightarrow 0 } \frac {\cos 5x -\cos x}{\cos 3x - \cos x }=...$
$ \lim_{x \rightarrow 0 } \frac {\cos 5x -\cos x}{\cos 3x - \cos x } \\ \lim_{x \rightarrow 0 } \frac {-2 \sin 3x \sin 2x}{-2 \sin 2x \sin x } =3$
Soal 12 $\lim_{x \rightarrow 0 } \frac {\sin 3x +\sin x}{\tan 3x }=... $
$\lim_{x \rightarrow 0 } \frac {\sin 3x +\sin x}{\tan 3x } \\ \lim_{x \rightarrow 0 } \frac {\sin 2x . \cos x}{\tan 3x } \\ \lim_{x \rightarrow 0 } \frac {3 . \cos x}{3 } = \frac {2}{3}$
Soal 13 . $\lim_{x \rightarrow 0 } \frac {\sin 2x -1}{\sin 10x -1 } =... $
Turunkan pembilang dan penyebut.
$\lim_{x \rightarrow 0 } \frac {\sin 2x -1}{\sin 10x -1 } \\ \lim_{x \rightarrow 0 } \frac { 2 \cos 2x}{10 \cos 10x } = \frac {1}{5}$
Catatan lain:
Anda harus ingat rumus penjumlahan trigonometri
$\lim_{x \rightarrow 0 } \frac {\tan ax }{\tan bx } \\ \lim_{x \rightarrow 0 } \frac {\tan ax }{\sin bx } \\ \lim_{x \rightarrow 0 } \frac {\sin ax }{\tan bx } \\ \lim_{x \rightarrow 0 } \frac {\sin ax }{bx } \\ \lim_{x \rightarrow 0 } \frac {\sin ax }{\sin bx } \\ \lim_{x \rightarrow 0 } \frac { ax }{\sin bx } \\ \text {Nilainya: } \frac {a}{b}$
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