Ujian Nasional - Pembahasan Soal UN Limit Fungsi Trigonometri - adalah | ciri-ciri | nama-nama

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

1. UN 2005
Nilai

_{x \to 0}^{l i m} \frac{s i n 3 x - s i n 3 x c o s 2 x}{2 x^{3}}

= ...
A.

\frac{1}{2}

\frac{2}{3}

\frac{3}{2}

D. 2
E. 3

Pembahasan :

_{x \to 0}^{l i m} \frac{s i n 3 x - s i n 3 x c o s 2 x}{2 x^{3}}

_{x \to 0}^{l i m} \frac{s i n 3 x (1 - c o s 2 x)}{2 x^{3}}

_{x \to 0}^{l i m} \frac{s i n 3 x (2 s i n^{2} x)}{2 x^{3}}

_{x \to 0}^{l i m} \frac{s i n 3 x s i n^{2} x}{x^{3}}

_{x \to 0}^{l i m} \frac{s i n 3 x}{x}

_{x \to 0}^{l i m} \frac{s i n x}{x}

_{x \to 0}^{l i m} \frac{s i n x}{x}

= 3 × 1 × 1 = 3

Jawaban : E

2. UN 2006
Nilai

_{x \to \frac{π}{4}}^{l i m} \frac{c o s 2 x}{c o s x - s i n x}

= ...
A. 0
B.

\frac{1}{2}

√2
C. 1
D. √2
E. ∞

Pembahasan :

_{x \to \frac{π}{4}}^{l i m} \frac{c o s 2 x}{c o s x - s i n x}

_{x \to \frac{π}{4}}^{l i m} \frac{c o s^{2} x - s i n^{2} x}{c o s x - s i n x}

_{x \to \frac{π}{4}}^{l i m} \frac{(c o s x - s i n x) (c o s x + s i n x)}{c o s x - s i n x}

_{x \to \frac{π}{4}}^{l i m} (c o s x + s i n x)

= cos

\frac{π}{4}

+ sin

\frac{π}{4}

\frac{1}{2}

√2 +

\frac{1}{2}

√2 = √2

Jawaban : D

3. UN 2007
Nilai

_{x \to 0}^{l i m} \frac{1 - c o s 2 x}{x t a n \frac{1}{2} x}

= ...
A. −4
B. −2
C. 1
D. 2
E. 4

Pembahasan :

_{x \to 0}^{l i m} \frac{1 - c o s 2 x}{x t a n \frac{1}{2} x}

_{x \to 0}^{l i m} \frac{2 s i n^{2} x}{x t a n \frac{1}{2} x}

2 ×

_{x \to 0}^{l i m} \frac{s i n x}{x}

_{x \to 0}^{l i m} \frac{s i n x}{t a n \frac{1}{2} x}

= 2 × 1 ×

\frac{1}{\frac{1}{2}}

= 4

Jawaban : E

4. UN 2009
Nilai

_{x \to \frac{π}{3}}^{l i m} \frac{t a n (3 x - π) c o s 2 x}{s i n (3 x - π)}

= ...
A.

- \frac{1}{2}

\frac{1}{2}

\frac{1}{2}

√2
D.

\frac{1}{2}

√3
E.

\frac{3}{2}

Pembahasan :

_{x \to \frac{π}{3}}^{l i m} \frac{t a n (3 x - π) c o s 2 x}{s i n (3 x - π)}

_{x \to \frac{π}{3}}^{l i m} c o s 2 x

_{x \to \frac{π}{3}}^{l i m} \frac{t a n (3 x - π)}{s i n (3 x - π)}

Misalkan u = 3x − π
Jika x →

\frac{π}{3}

maka u → 0

_{x \to \frac{π}{3}}^{l i m} c o s 2 x

_{u \to 0}^{l i m} \frac{t a n u}{s i n u}

= cos 2(

\frac{π}{3}

) × 1

= cos (

\frac{2 π}{3}

) =

- \frac{1}{2}

Jawaban : A

5. UN 2010
Nilai

_{x \to 0}^{l i m} (\frac{c o s 4 x s i n 3 x}{5 x})

= ...
A.

\frac{5}{3}

B. 1
C.

\frac{3}{5}

\frac{1}{5}

E. 0

Pembahasan :

_{x \to 0}^{l i m} \frac{c o s 4 x s i n 3 x}{5 x}

_{x \to 0}^{l i m} c o s 4 x

_{x \to 0}^{l i m} \frac{s i n 3 x}{5 x}

= cos (4.0) ×

\frac{3}{5}

= 1 ×

\frac{3}{5}

\frac{3}{5}

Jawaban : C

6. UN 2010
Nilai

_{x \to 0}^{l i m} (\frac{s i n 4 x - s i n 2 x}{6 x})

= ...
A. 1
B.

\frac{2}{3}

\frac{1}{2}

\frac{1}{3}

\frac{1}{6}

Pembahasan :

_{x \to 0}^{l i m} \frac{s i n 4 x - s i n 2 x}{6 x}

_{x \to 0}^{l i m} \frac{2 c o s \frac{1}{2} (4 x + 2 x) s i n \frac{1}{2} (4 x - 2 x)}{6 x}

\frac{2}{6}

_{x \to 0}^{l i m} \frac{c o s 3 x s i n x}{x}

\frac{2}{6}

_{x \to 0}^{l i m}

cos 3x ×

_{x \to 0}^{l i m} \frac{s i n x}{x}

\frac{2}{6}

× cos (3.0) × 1 =

\frac{1}{3}

Jawaban : D

7. UN 2011
Nilai

_{x \to 0}^{l i m} \frac{1 - c o s 2 x}{2 x s i n 2 x}

= ...
A.

\frac{1}{8}

\frac{1}{6}

\frac{1}{4}

\frac{1}{2}

E. 1

Pembahasan :

_{x \to 0}^{l i m} \frac{1 - c o s 2 x}{2 x s i n 2 x}

_{x \to 0}^{l i m} \frac{2 s i n^{2} x}{2 x s i n 2 x}

_{x \to 0}^{l i m} \frac{s i n x}{x}

_{x \to 0}^{l i m} \frac{s i n x}{s i n 2 x}

= 1 ×

\frac{1}{2}

\frac{1}{2}

Jawaban : D

8. UN 2012
Nilai

_{x \to 0}^{l i m} \frac{c o s 4 x - 1}{x t a n 2 x}

= ...
A. 4
B. 2
C. −1
D. −2
E. −4

Pembahasan :

_{x \to 0}^{l i m} \frac{- (1 - c o s 4 x)}{x t a n 2 x}

_{x \to 0}^{l i m} \frac{- (2 s i n^{2} 2 x)}{x t a n 2 x}

−2 ×

_{x \to 0}^{l i m} \frac{s i n 2 x}{x} \times_{x \to 0}^{l i m} \frac{s i n 2 x}{t a n 2 x}

= −2 ×

\frac{2}{1}

\frac{2}{2}

= −4

Jawaban : E

9. UN 2013
Nilai dari

_{x \to - 2}^{l i m} \frac{(x^{2} - 4) t a n (x + 2)}{s i n^{2} (x + 2)} = . . .

A. −4
B. −3
C. 0
D. 4
E. ∞

Pembahasan :

_{x \to - 2}^{l i m} \frac{(x - 2) (x + 2) t a n (x + 2)}{s i n^{2} (x + 2)}

_{x \to - 2}^{l i m}

(x − 2) ×

_{x \to - 2}^{l i m} \frac{(x + 2)}{s i n (x + 2)}

\frac{t a n (x + 2)}{s i n (x + 2)}

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

_{x \to - 2}^{l i m}

(x − 2) ×

_{u \to 0}^{l i m} \frac{u}{s i n u}

_{u \to 0}^{l i m} \frac{t a n u}{s i n u}

= (−2 − 2) × 1 × 1 = −4

Jawaban : A

10. UN 2013
Nilai dari

_{x \to 3}^{l i m} \frac{x t a n (2 x - 6)}{s i n (x - 3)} = . . .

A. 0
B.

\frac{1}{2}

C. 2
D. 3
E. 6

Pembahasan :

_{x \to 3}^{l i m} \frac{x t a n 2 (x - 3)}{s i n (x - 3)}

_{x \to 3}^{l i m} x \times_{x \to 3}^{l i m} \frac{t a n 2 (x - 3)}{s i n (x - 3)}

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

_{x \to 3}^{l i m} x \times_{u \to 0}^{l i m} \frac{t a n 2 u}{s i n u}

= 3 × 2 = 6

Jawaban : E

11. UN 2014
Nilai

_{x \to 0}^{l i m} \frac{1 - c o s 8 x}{s i n 2 x t a n 2 x}

= ...
A. 16
B. 12
C. 8
D. 4
E. 2

Pembahasan :

_{x \to 0}^{l i m} \frac{1 - c o s 8 x}{s i n 2 x t a n 2 x}

_{x \to 0}^{l i m} \frac{2 s i n^{2} 4 x}{s i n 2 x t a n 2 x}

2 ×

_{x \to 0}^{l i m} \frac{s i n 4 x}{s i n 2 x} \times_{x \to 0}^{l i m} \frac{s i n 4 x}{t a n 2 x}

= 2 ×

\frac{4}{2}

\frac{4}{2}

= 8

Jawaban : C

12. UN 2015
Nilai dari

_{x \to 0}^{l i m} \frac{x t a n x}{2 c o s^{2} x - 2}

= ...
A.

- \frac{1}{2}

- \frac{1}{4}

C. 0
D.

\frac{1}{2}

E. 1

Pembahasan :

_{x \to 0}^{l i m} \frac{x t a n x}{2 c o s^{2} x - 2}

_{x \to 0}^{l i m} \frac{x t a n x}{- 2 (1 - c o s^{2} x)}

_{x \to 0}^{l i m} \frac{x t a n x}{- 2 s i n^{2} x}

- \frac{1}{2}

_{x \to 0}^{l i m} \frac{x}{s i n x} \times_{x \to 0}^{l i m} \frac{t a n x}{s i n x}

- \frac{1}{2}

× 1 × 1 =

- \frac{1}{2}

Jawaban : A

13. UN 2015
Nilai

_{x \to 0}^{l i m} \frac{x t a n 3 x}{1 - c o s^{2} 2 x}

= ...
A. 0
B.

\frac{1}{4}

\frac{2}{4}

\frac{3}{4}

E. 1

Pembahasan :

_{x \to 0}^{l i m} \frac{x t a n 3 x}{1 - c o s^{2} 2 x}

_{x \to 0}^{l i m} \frac{x t a n 3 x}{s i n^{2} 2 x}

_{x \to 0}^{l i m} \frac{x}{s i n 2 x} \times_{x \to 0}^{l i m} \frac{t a n 3 x}{s i n 2 x}

\frac{1}{2}

\frac{3}{2}

\frac{3}{4}

Jawaban : D

14. UN 2016
Nilai

_{x \to 0}^{l i m} \frac{1 - c o s 4 x}{2 x s i n 4 x}

= ...
A. 1
B.

\frac{1}{2}

C. 0
D.

- \frac{1}{2}

E. −1

Pembahasan :

\begin{aligned} lim_{x \to 0} \frac{1 - c o s 4 x}{2 x s i n 4 x} & = lim_{x \to 0} \frac{⧸ 2 s i n^{2} 2 x}{⧸ 2 x s i n 4 x} \\ = lim_{x \to 0} (\frac{s i n 2 x}{x} \cdot \frac{s i n 2 x}{s i n 4 x}) \\ = \frac{2}{1} \cdot \frac{2}{4} \\ = 1 \end{aligned}

Jawaban : A

sumber: https://smatika.blogspot.com/

Ujian Nasional - Pembahasan Soal UN Limit Fungsi Trigonometri

sumber: https://smatika.blogspot.com/

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