Limit and trigonometry

Calculus Level 3

E = 2016 sin ⁡ x , M = 2016 cos ⁡ x , T = 2016 tan ⁡ x \large E =\frac{2016}{\sin{x}}, \qquad M =\frac{2016}{\cos{x}}, \qquad T =\frac{2016}{\tan{x}}

lim ⁡ x → 0 + E M T = ? \large \lim_{x\rightarrow0^{+}} EMT = \, ?

− ∞ -\infty 0 1 ∞ \infty Undefined

Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Tommy Li
Aug 1, 2016

E = 2016 sin ⁡ x , M = 2016 cos ⁡ x , T = 2016 tan ⁡ x \large E =\frac{2016}{\sin{x}} , M =\frac{2016}{\cos{x}} , T =\frac{2016}{\tan{x}}

lim ⁡ x → 0 + E M T = lim ⁡ x → 0 + 201 6 3 sin ⁡ x cos ⁡ x tan ⁡ x \large \lim_{x\rightarrow0^{+}} EMT = \lim_{x\rightarrow0^{+}}\dfrac{2016^3}{\sin{x}\cos{x}\tan{x}}

lim ⁡ x → 0 + E M T → 201 6 3 ( 0 + ) ( 1 ) ( 0 + ) \large \lim_{x\rightarrow0^{+}} EMT \rightarrow \dfrac{2016^3}{(0^{+})(1)(0^{+})}

lim ⁡ x → 0 + E M T → 201 6 3 0 + \large \lim_{x\rightarrow0^{+}} EMT \rightarrow \dfrac{2016^3}{0^{+}}

lim ⁡ x → 0 + E M T → ∞ \large \lim_{x\rightarrow0^{+}} EMT \rightarrow \infty

lim ⁡ x → 0 + E M T = lim ⁡ x → 0 + 201 6 3 sin ⁡ x cos ⁡ x tan ⁡ x \large \lim_{x\rightarrow0^{+}} EMT = \lim_{x\rightarrow0^{+}}\dfrac{2016^3}{\sin{x}\cos{x}\tan{x}}

lim ⁡ x → 0 + E M T = lim ⁡ x → 0 + 201 6 3 sin ⁡ x cos ⁡ x sin ⁡ x cos ⁡ x \large \lim_{x\rightarrow0^{+}} EMT = \lim_{x\rightarrow0^{+}}\dfrac{2016^3}{\sin{x}\cos{x}\frac{\sin{x}}{\cos{x}}}

lim ⁡ x → 0 + E M T = lim ⁡ x → 0 + 201 6 3 sin ⁡ 2 x \large \lim_{x\rightarrow0^{+}} EMT = \lim_{x\rightarrow0^{+}}\dfrac{2016^3}{\sin^2{x}}

lim ⁡ x → 0 + E M T → 201 6 3 ( 0 + ) 2 \large \lim_{x\rightarrow0^{+}} EMT \rightarrow \dfrac{2016^3}{(0^{+})^2}

lim ⁡ x → 0 + E M T → 201 6 3 0 + \large \lim_{x\rightarrow0^{+}} EMT \rightarrow \dfrac{2016^3}{0^{+}}

lim ⁡ x → 0 + E M T → ∞ \large \lim_{x\rightarrow0^{+}} EMT \rightarrow \infty

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