Pembahasan Soal Un Limit Fungsi Trigonometri

Pembahasan soal-soal Ujian Nasional (UN) Sekolah Menengan Atas bidang studi Matematika IPA untuk pokok bahasan Limit Fungsi Trigonometri.

1. UN 2005
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{sin\,3x-sin\,3x\:cos\,2x}{2x^{3}}}\) = ...
A.  \(\frac{1}{2}\)
B.  \(\frac{2}{3}\)
C.  \(\frac{3}{2}\)
D.  2
E.  3

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{sin\,3x-sin\,3x\:cos\,2x}{2x^{3}}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{sin\,3x(1-cos\,2x)}{2x^{3}}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{sin\,3x\,(2\,sin^{2}x)}{2x^{3}}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{sin\,3x\,sin^{2}x}{x^{3}}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{sin\,3x}{x}}\) × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,x}{x}}\) × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,x}{x}}\)

= 3 × 1 × 1 = 3

Jawaban : E


2. UN 2006
Nilai \(\mathrm{_{x \to \frac{\pi }{4}}^{lim}\frac{cos\,2x}{cos\,x-sin\,x}}\) = ...
A.  0
B.  \(\frac{1}{2}\)√2
C.  1
D.  √2
E.  ∞

Pembahasan :
\(\mathrm{_{x \to \frac{\pi }{4}}^{lim}\frac{cos\,2x}{cos\,x-sin\,x}}\)

\(\mathrm{_{x \to \frac{\pi }{4}}^{lim}\frac{cos^{2}x-sin^{2}x}{cos\,x-sin\,x}}\)

\(\mathrm{_{x \to \frac{\pi }{4}}^{lim}\frac{(cos\,x-sin\,x)(cos\,x+sin\,x)}{cos\,x-sin\,x}}\)

\(\mathrm{_{x \to \frac{\pi }{4}}^{lim}(cos\,x+sin\,x)}\)

= cos \(\frac{\pi}{4}\) + sin \(\frac{\pi}{4}\)

= \(\frac{1}{2}\)√2 + \(\frac{1}{2}\)√2 = √2

Jawaban : D



3. UN 2007
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,2x}{x\,tan\,\frac{1}{2}x}}\) = ...
A.  −4
B.  −2
C.  1
D.  2
E.  4

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,2x}{x\,tan\,\frac{1}{2}x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{2\,sin^{2}x}{x\,tan\,\frac{1}{2}x}}\)

2 × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,x}{x}}\) × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,x}{tan\,\frac{1}{2}x}}\)

= 2 × 1 × \(\frac{1}{\frac{1}{2}}\) = 4

Jawaban : E


4. UN 2009
Nilai \(\mathrm{_{x \to \frac{\pi }{3}}^{lim}\frac{tan(3x-\pi )\,cos\,2x}{sin(3x-\pi )}}\) = ...
A.  \(-\frac{1}{2}\)
B.  \(\frac{1}{2}\)
C.  \(\frac{1}{2}\)√2
D.  \(\frac{1}{2}\)√3
E.  \(\frac{3}{2}\)

Pembahasan :
\(\mathrm{_{x \to \frac{\pi }{3}}^{lim}\frac{tan(3x-\pi )\,cos\,2x}{sin(3x-\pi )}}\)

\(\mathrm{_{x \to \frac{\pi }{3}}^{lim}cos\,2x}\) × \(\mathrm{_{x \to \frac{\pi }{3}}^{lim}\frac{tan(3x-\pi )}{sin(3x-\pi )}}\)

Misalkan u = 3x − π
Jika x → \(\frac{\pi}{3}\) maka u → 0

\(\mathrm{_{x \to \frac{\pi }{3}}^{lim}cos\,2x}\) × \(\mathrm{_{u \to 0}^{lim}\frac{tan\,u}{sin\,u}}\)

= cos 2(\(\frac{\pi}{3}\)) × 1

= cos (\(\frac{2\pi}{3}\)) = \(-\frac{1}{2}\)

Jawaban : A



5. UN 2010
Nilai \(\mathrm{_{x \to 0}^{lim}\left ( \frac{cos\,4x\,sin\,3x}{5x} \right )}\) = ...
A.  \(\frac{5}{3}\)
B.  1
C.  \(\frac{3}{5}\)
D.  \(\frac{1}{5}\)
E.  0

Pembahasan :
\(\mathrm{_{x \to 0}^{lim} \frac{cos\,4x\,sin\,3x}{5x}}\)

\(\mathrm{_{x \to 0}^{lim} cos\,4x}\) × \(\mathrm{_{x \to 0}^{lim} \frac{sin\,3x}{5x}}\)

= cos (4.0) × \(\frac{3}{5}\)

= 1 × \(\frac{3}{5}\) = \(\frac{3}{5}\)

Jawaban : C


6. UN 2010
Nilai \(\mathrm{_{x \to 0}^{lim}\left ( \frac{sin\,4x-sin\,2x}{6x} \right )}\) = ...
A.  1
B.  \(\frac{2}{3}\)
C.  \(\frac{1}{2}\)
D.  \(\frac{1}{3}\)
E.  \(\frac{1}{6}\)

Pembahasan :
\(\mathrm{_{x \to 0}^{lim} \frac{sin\,4x-sin\,2x}{6x}}\)

\(\mathrm{_{x \to 0}^{lim} \frac{2\,cos\,\frac{1}{2}(4x+2x)\,sin\,\frac{1}{2}(4x-2x)}{6x} }\)

\(\frac{2}{6}\) × \(\mathrm{_{x \to 0}^{lim} \frac{cos\,3x\,sin\,x}{x}}\)

\(\frac{2}{6}\) × \(\mathrm{_{x \to 0}^{lim}}\) cos 3x × \(\mathrm{_{x \to 0}^{lim} \frac{sin\,x}{x}}\)

= \(\frac{2}{6}\) × cos (3.0) × 1 = \(\frac{1}{3}\)

Jawaban : D


7. UN 2011
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,2x}{2x\,sin\,2x}}\) = ...
A.  \(\frac{1}{8}\)
B.  \(\frac{1}{6}\)
C.  \(\frac{1}{4}\)
D.  \(\frac{1}{2}\)
E.  1

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,2x}{2x\,sin\,2x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{2\,sin^{2}x}{2x\,sin\,2x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{sin\,x}{x}}\) × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,x}{sin\,2x}}\)

= 1 × \(\frac{1}{2}\) = \(\frac{1}{2}\)

Jawaban : D


8. UN 2012
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{cos\,4x-1}{x\,tan\,2x}}\) = ...
A.  4
B.  2
C.  −1
D.  −2
E.  −4

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{-(1-cos\,4x)}{x\,tan\,2x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{-(2sin^{2}2x)}{x\,tan\,2x}}\)

−2 × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,2x}{x}\;\times\;_{x \to 0}^{lim}\frac{sin\,2x}{tan\,2x} }\)

= −2 × \(\frac{2}{1}\) × \(\frac{2}{2}\) = −4

Jawaban : E


9. UN 2013
Nilai dari \(\mathrm{_{x \to -2}^{lim}\frac{(x^{2}-4)\;tan\,(x+2)}{sin^{2}(x+2)}=...}\)
A.  −4
B.  −3
C.  0
D.  4
E.  ∞

Pembahasan :
\(\mathrm{_{x \to -2}^{lim}\frac{(x-2)(x+2)\;tan\,(x+2)}{sin^{2}(x+2)}}\)

\(\mathrm{_{x \to -2}^{lim}}\) (x − 2) × \(\mathrm{_{x \to -2}^{lim}\frac{(x+2)}{sin(x+2)}}\)×\(\mathrm{\frac{tan(x+2)}{sin(x+2)}}\)

Misalkan u = x + 2
Jika x → −2 maka u → 0

\(\mathrm{_{x \to -2}^{lim}}\) (x − 2) × \(\mathrm{_{u \to 0}^{lim}\frac{u}{sin\,u}}\) × \(\mathrm{_{u \to 0}^{lim}\frac{tan\,u}{sin\,u}}\)

= (−2 − 2) × 1 × 1 = −4

Jawaban : A


10. UN 2013
Nilai dari \(\mathrm{_{x \to 3}^{lim}\frac{x\,tan\,(2x-6)}{sin\,(x-3)}=...}\)
A.  0
B.  \(\frac{1}{2}\)
C.  2
D.  3
E.  6

Pembahasan :
\(\mathrm{_{x \to 3}^{lim}\frac{x\,tan\,2(x-3)}{sin\,(x-3)}}\)

\(\mathrm{_{x \to 3}^{lim}\,x\;\times\; _{x \to 3}^{lim}\frac{tan\,2(x-3)}{sin\,(x-3)}}\)

Misalkan u = x − 3
Jika x → 3 maka u → 0

\(\mathrm{_{x \to 3}^{lim}\,x\;\times\; _{u \to 0}^{lim}\frac{tan\,2u}{sin\,u}}\)

= 3 × 2 = 6

Jawaban : E


11. UN 2014
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,8x}{sin\,2x\,tan\,2x}}\) = ...
A.  16
B.  12
C.  8
D.  4
E.  2

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,8x}{sin\,2x\,tan\,2x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{2\,sin^{2}4x}{sin\,2x\,tan\,2x}}\)

2 × \(\mathrm{_{x \to 0}^{lim}\frac{sin\,4x}{sin\,2x}\;\times\;_{x \to 0}^{lim}\frac{sin\,4x}{tan\,2x} }\)

= 2 × \(\frac{4}{2}\) × \(\frac{4}{2}\) = 8

Jawaban : C


12. UN 2015
Nilai dari \(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,x}{2\,cos^{2}x\,-2}}\) = ...
A.  \(-\frac{1}{2}\)
B.  \(-\frac{1}{4}\)
C.  0
D.  \(\frac{1}{2}\)
E.  1

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,x}{2\,cos^{2}x\,-2}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,x}{-2(1-cos^{2}x)}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,x}{-2\,sin^{2}x}}\)

 \(-\frac{1}{2}\) × \(\mathrm{_{x \to 0}^{lim}\frac{x}{sin\,x}\;\times\;_{x \to 0}^{lim}\frac{tan\,x}{sin\,x} }\)

=  \(-\frac{1}{2}\) × 1 × 1 =  \(-\frac{1}{2}\)

Jawaban : A


13. UN 2015
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,3x}{1-cos^{2}2x}}\) = ...
A.  0
B.  \(\frac{1}{4}\)
C.  \(\frac{2}{4}\)
D.  \(\frac{3}{4}\)
E.  1

Pembahasan :
\(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,3x}{1-cos^{2}2x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{x\,tan\,3x}{sin^{2}2x}}\)

\(\mathrm{_{x \to 0}^{lim}\frac{x}{sin\,2x}\;\times\;_{x \to 0}^{lim}\frac{tan\,3x}{sin\,2x} }\)

= \(\frac{1}{2}\) × \(\frac{3}{2}\) = \(\frac{3}{4}\)

Jawaban : D


14. UN 2016
Nilai \(\mathrm{_{x \to 0}^{lim}\frac{1-cos\,4x}{2x\,sin\,4x}}\) = ...
A.  1
B.  \(\frac{1}{2}\)
C.  0
D.  \(-\frac{1}{2}\)
E.  −1

Pembahasan :
\(\begin{align}
\mathrm{\lim_{x \to 0}\,\frac{1-cos\,4x}{2x\,sin\,4x}}
& = \mathrm{\lim_{x \to 0}\,\frac{{\color{red}\not}2\,sin^{2}2x}{{\color{red}\not}2x\,sin\,4x}} \\
& = \mathrm{\lim_{x \to 0}\,\left (\frac{sin\,2x}{x}\cdot \frac{sin\,2x}{sin\,4x}  \right )} \\
& = \frac{2}{1}\cdot \frac{2}{4} \\
& = 1
\end{align}\)

Jawaban : A



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