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Selamat datang di halaman contoh soal dan pembahasan "Limit Trigonometri ". Di halaman ini akan membahas tentang contoh soal dan pembahasan lengkap mengenai Limit Trigonometri.
Untuk lebih jelasnya mari kita lihat contoh soal dan pembahasan dibawah ini!
1. \(\lim\limits_{x\rightarrow 0}\frac{sin3x}{2x} \)=
a.\(\frac{1}{2}\)
b.\(\frac{1}{3}\)
c.\(\frac{3}{2}\)
d.\(\frac{4}{2}\)
e.\(\frac{6}{3}\)
pembahasan :
\(\lim\limits_{x\rightarrow0}\frac{sin3x}{2x}\)
=\(\lim\limits_{x\rightarrow0}\frac{sin3x}{2x}.\frac{3x}{3x}\)
=\(\lim\limits_{x\rightarrow0}\frac{sin3x}{3x}.\frac{3x}{2x}\)
=\(1.\frac{3}{2}\)
=\(\frac{3}{2}\)
2.\(\lim\limits_{x\rightarrow 0}\frac{5x}{3sin3x}\)=
a.\(\frac{1}{2}\)
b.\(\frac{5}{2}\)
c.\(\frac{1}{3}\)
d.\(\frac{7}{3}\)
e.\(\frac{5}{9}\)
pembahasan:
\(\lim\limits_{x\rightarrow 0}\frac{5x}{3sin3x}\)
=\(\lim\limits_{x\rightarrow 0}\frac{5x}{3sin3x}.\frac{3x}{3x}\)
=\(\lim\limits_{x\rightarrow 0}\frac{3x}{3sin3x}.\frac{5x}{3x}\)
=\(\lim\limits_{x\rightarrow 0}\frac{1}{3}.\frac{3x}{sin3x}.\frac{5x}{3x}\)
=\(\frac{1}{3}.1\frac{5}{3}\)
=\(\frac{5}{9}\)
3.\(\lim\limits_{x\rightarrow 0} \frac{sin4x}{3x}\)=
a.\(\frac{1}{2}\)
b.\(\frac{2}{2}\)
c.\(\frac{3}{4}\)
d.\(\frac{2}{3}\)
e.\(\frac{4}{3}\)
pembahasan
\(\Rightarrow \frac{sin ax}{bx}\)=\(\frac{a}{b}\)=\(\frac{sin 4x}{3x}\)=\(\frac{4}{3}\)
4. \(\lim\limits_{x\rightarrow 0} \frac{sin2x}{sin3x}\)=
a.\(\frac{1}{2}\)
b.\(\frac{3}{2}\)
c.\(\frac{2}{3}\)
d.\(\frac{6}{5}\)
e.\(\frac{3}{4}\)
pembahasan:
\(\Rightarrow \frac{sin ax}{sinbx}\)=\(\frac{a}{b}\)
=\(\frac{sin 2x}{sin3x}=\frac{2}{3}\)
5. \(\lim\limits_{x\rightarrow 0} \frac{sin2x}{tan7x}\)=
a.\(\frac{3}{4}\)
b.\(\frac{3}{5}\)
c.\(\frac{4}{3}\)
d.\(\frac{3}{7}\)
e.\(\frac{2}{7}\)
pembahasan:
\(\Rightarrow \frac{sin ax}{tanbx}\)=\(\frac{a}{b}\)
=\(\frac{sin 2x}{tan7x}\)=\(\frac{2}{7}\)
6. \(\lim\limits_{x\rightarrow \frac{1}{2}} \frac{sin(4x-2)}{tan(2x-1)}\)=
a.\(2\)
b.\(3\)
c.\(5\)
d.\(4\)
e.\(6\)
pembahasan:
misalkan: \(a=2x-1\)\(x=\frac{1}{2}\),
\(a=\lim\limits_{x\rightarrow \frac{1}{2}} \frac{sin(4x-2)}{tan(2x-1)}\)=\(\lim\limits_{x\rightarrow \frac{1}{2}} \frac{sin2(2x-1)}{tan(2x-1)}\)=
\(\lim\limits_{x\rightarrow \frac{1}{2}} \frac{sin2(a)}{tan(a)}\)=\(2\)
7.\(\lim\limits_{x\rightarrow 0} \frac{x^{2}+ \sin{x} \tan{x}} {1-cos2x}\)=
a.\(1\)
b.\(2\)
c.\(3\)
d.\(4\)
e.\(5\)
pembahasan:
\(\lim\limits_{x\rightarrow 0}\frac{x^{2}+\sin{x}\tan{x}}{1-(1-2\sin^2x)}\)=
\(\lim\limits_{x\rightarrow 0}\frac{x^{2}+\sin{x}\tan{x}}{2\sin^2x}\)=
\(\lim\limits_{x\rightarrow 0}\frac{x^{2}}{2\sin^{2}x}+\frac{\sin x\tan x}{2\sin^{2}}\)=
\(\lim\limits_{x\rightarrow 0}\frac{1}{2}.\frac{x}{sinx}.\frac{x}{sinx}+\frac{1}{2}.\frac{sinx}{sinx}.\frac{tanx}{sinx}\)=
\(\frac{1}{2}.1.1+\frac{1}{2}.1.1=1\)
8. \(\lim\limits_{x\rightarrow 0}\frac{2-2\cos2x}{x^2}\)=
a.\(1\)
b.\(3\)
c.\(4\)
d.\(5\)
e.\(6\)
pembahsan :
\(\lim\limits_{x\rightarrow 0} \frac{2-2\cos2x}{x^2}\)
=\(\lim\limits_{x\rightarrow 0} \frac{2(\cos2x)}{x^2}\)
=\(\lim\limits_{x\rightarrow 0} \frac{2(1-(1-2\sin^{2}x)}{x^2}\)
=\(\lim\limits_{x\rightarrow 0}\frac{2(1-1+2\sin^{2}x)}{x^2}\)
=\(\lim\limits_{x\rightarrow 0}\frac{2(2sin^{2}x)}{x^2}\)
=\(\lim\limits_{x\rightarrow 0}\frac{(4sin^{2}x)}{x^2}\)
=\(4.\lim\limits_{x\rightarrow 0}\frac{(sinx)^{2}}{x}\)
=\(4.1^{2}\)=\(4\)
9. \(\lim\limits_{x\rightarrow 0} \frac{x\tan3x}{sin^2 6x}\)=
a.\(\frac{1}{3}\)
b.\(\frac{1}{4}\)
c.\(\frac{1}{15}\)
d.\(\frac{1}{7}\)
e.\(\frac{1}{12}\)
pembahasan :
\(\lim\limits_{x\rightarrow 0} \frac{x\tan3x}{sin^2 6x}\)=
\(\frac{x\tan3x}{sin6x.sin6x}\)=\(\frac{1x}{sin6x}.\frac{tan3x}{sin6x}\)
=\(\frac{1.3}{6.6}\)=\(\frac{1}{2}\)
10. \(\lim\limits_{x\rightarrow 0} \frac{\cos4x-1}{x\tan2x}\)=
a.\(1\)
b.\(-1\)
c.\(-4\)
d.\(4\)
e.\(0\)
pembahasan :
\(\lim\limits_{x\rightarrow 0} \frac{\cos4x-1}{x\tan2x}\)=
\(\frac{-2sin^{2}2x}{x\tan2x}\)=\(\frac{-2sin2x}{x}.\frac{sin2x}{tan2x}\)
=\(\frac{-2.2.2}{1.2}\)=\(-4\)
