Question

Hasil dari [tex]\lim_{x \to 0} (\text{csc}x-\frac{1}{x} )[/tex] adalah ...
A. 4 B. 3 C. 2 D. 1 E. 0

Answer 1

Hasil dari [tex]\lim\limits_{x \to 0} (\text{csc}\: x-\frac{1}{x} )[/tex] adalah E. 0Pembahasan:[tex] \small \boxed{\begin{aligned}\sf \lim_{x \to 0} (\text{csc} \: x-\frac{1}{x} )&=\sf \lim_{x \to 0}\left(\dfrac{1 }{sin \: x} - \frac{1}{x} \right)\\ \sf &=\sf \lim_{x \to 0}\left(\dfrac{ x - sin \: x}{x \: sin \: x} \right)\\ \sf &= \sf \lim_{x \to 0}\left(\dfrac{ x - tan \: x \: cos \: x}{x \: sin \: x} \right)\\ \sf &=\sf\lim_{x \to 0}\left( \frac{x}{x \: sin \: x} - \dfrac{ tan \: x \: cos \: x}{x \: sin \: x} \right)\\ \sf &=\sf \lim_{x \to 0}\left(\dfrac{ 1}{x} - \frac{cos \: x}{x} \right)\\ \sf &= \sf \lim_{x \to 0}\left(\dfrac{1 - cos \: x }{x} \right)\\ \sf &=\sf \lim_{x \to 0}\left(\dfrac{1 - (1 - 2 \: {sin}^{2}( \frac{1}{2} x))}{x} \right)\\ \sf &=\sf \lim_{x \to 0}\left(\dfrac{ {2 \: sin}^{2} ( \frac{1}{2})x}{x} \right)\\ \sf &= \sf \lim_{x \to 0}\left(\dfrac{ 2 \: sin \: \frac{1}{2} x \: \: sin \: \frac{1}{2} x}{x} \right)\\ \sf &=\sf\lim_{x \to 0}2. \frac{1}{2} \: sin \: \frac{1}{2}x \\ \sf &=\sf \: 2.\frac{1}{2} \: sin \: \frac{1}{2}(0) \\ \sf &=\sf 1 \: (sin \: 0 )\\ \sf &= \sf sin \: {0}^{ \circ} \\ \sf &=\sf0 \end{aligned}}[/tex]Jadi, opsi yang tepat adalah (E).