Pertanyaan

Tentukan nilai limit fungsi berikut. b. x → 0 lim 2 3 x sin ( 4 x ) 8 tan 2 ( 2 1 x )

Tentukan nilai limit fungsi berikut.

b. $x \to 0 lim \frac{8 tan ^{2} ( \frac{1}{2} x )}{\frac{3}{2} x sin ( 4 x )}$

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Habis dalam

Klaim

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

lim x → 0 2 3 x sin ( 4 x ) 8 t a n 2 ( 2 1 x ) = 3 4 .

lim_{x \to 0} \frac{8 t a n ^{2} ( \frac{1}{2} x )}{\frac{3}{2} x sin ( 4 x )} = \frac{4}{3} .

Pembahasan

Substitusi langsung: lim x → 0 2 3 x sin ( 4 x ) 8 t a n 2 ( 2 1 x ) = = 0 ⋅ sin ( 0 ) 8 t a n 2 ( 0 ) 0 0 ( tak tentu ) Mengubah fungsi trigonometri: lim x → 0 2 3 x sin ( 4 x ) 8 t a n 2 ( 2 1 x ) = = = 2 3 8 ⋅ lim x → 0 x t a n ( 2 1 x ) ⋅ lim x → 0 sin ( 4 x ) t a n ( 2 1 x ) 3 16 ⋅ 2 1 ⋅ 2 1 3 4 Dengan demikian, lim x → 0 2 3 x sin ( 4 x ) 8 t a n 2 ( 2 1 x ) = 3 4 .

Substitusi langsung:

$lim_{x \to 0} \frac{8 t a n ^{2} ( \frac{1}{2} x )}{\frac{3}{2} x sin ( 4 x )} = = \frac{8 t a n ^{2} ( 0 )}{0 \cdot sin ( 0 )} \frac{0}{0} (tak tentu)$

Mengubah fungsi trigonometri:

$lim_{x \to 0} \frac{8 t a n ^{2} ( \frac{1}{2} x )}{\frac{3}{2} x sin ( 4 x )} = = = \frac{8}{\frac{3}{2}} \cdot lim_{x \to 0} \frac{t a n ( \frac{1}{2} x )}{x} \cdot lim_{x \to 0} \frac{t a n ( \frac{1}{2} x )}{sin ( 4 x )} \frac{16}{3} \cdot \frac{1}{2} \cdot \frac{1}{2} \frac{4}{3}$