Advertisements
Advertisements
Question
\[\lim_{x \to 0} \frac{2 \sin x^\circ - \sin 2 x^\circ}{x^3}\]
Solution
\[\lim_{x \to 0} \left[ \frac{2 \sin x° - \sin \left( 2x° \right)}{x^3} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin \left( \frac{\pi x}{180} \right) - \sin \left( \frac{2\pi x}{180} \right)}{x^3} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin \left( \frac{\pi x}{180} \right) - 2 \sin \left( \frac{\pi x}{180} \right) \times \cos\left( \frac{\pi x}{180} \right)}{x^3} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin \left( \frac{\pi x}{180} \right) \left[ 1 - \cos \left( \frac{\pi x}{180} \right) \right]}{x^3} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin \left( \frac{\pi x}{180} \right) \times 2 \sin^2 \left( \frac{\pi x}{360} \right)}{x \times x^2} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{4 \sin \left[ \frac{\pi x}{180} \right] \times \sin^2 \left[ \frac{\pi x}{360} \right]}{\frac{\pi x}{180} \times \frac{\pi x}{360} \times \frac{\pi x}{360}} \times \frac{\pi}{180} \times \left( \frac{\pi}{360} \right)^2 \right]\]
\[ = 4 \lim_{x \to 0} \left[ \frac{\sin \left( \frac{\pi x}{180} \right)}{\frac{\pi x}{180}} \times \frac{\sin \left( \frac{\pi x}{360} \right) \times \sin \left( \frac{\pi x}{360} \right)}{\frac{\pi x}{360} \times \frac{\pi x}{360}} \times \frac{\pi^3}{180 \times {360}^2} \right]\]
\[ = 4 \times 1 \times 1 \times 1 \times \frac{\pi^3}{180 \times 360 \times 360}\]
\[ = \left( \frac{\pi}{180} \right)^3\]
APPEARS IN
RELATED QUESTIONS
\[\lim_{x \to 0} 9\]
\[\lim_{x \to - 1} \frac{x^3 - 3x + 1}{x - 1}\]
\[\lim_{x \to 3} \frac{x^2 - 9}{x + 2}\]
\[\lim_{x \to 0} \frac{ax + b}{cx + d}, d \neq 0\]
\[\lim_{x \to 4} \frac{x^2 - 7x + 12}{x^2 - 3x - 4}\]
\[\lim_{x \to 2} \frac{x^4 - 16}{x - 2}\]
\[\lim_{x \to - 1} \frac{x^3 + 1}{x + 1}\]
\[\lim_{x \to \infty} \sqrt{x + 1} - \sqrt{x}\]
\[\lim_{x \to \infty} \sqrt{x^2 + 7x - x}\]
\[\lim_{n \to \infty} \left[ \frac{1 + 2 + 3 . . . . . . n - 1}{n^2} \right]\]
\[\lim_{x \to \infty} \left[ \sqrt{x}\left\{ \sqrt{x + 1} - \sqrt{x} \right\} \right]\]
\[\lim_{x \to 0} \frac{\sin x^0}{x}\]
\[\lim_{x \to 0} \frac{\sin x \cos x}{3x}\]
\[\lim_{x \to 0} \frac{\tan mx}{\tan nx}\]
\[\lim_{x \to 0} \frac{\sin x^n}{x^n}\]
\[\lim_{x \to 0} \frac{\tan^2 3x}{x^2}\]
\[\lim_{x \to 0} \frac{x \cos x + 2 \sin x}{x^2 + \tan x}\]
\[\lim_{x \to 0} \frac{\tan x - \sin x}{\sin 3x - 3 \sin x}\]
\[\lim_{x \to 0} \frac{\sin 2x \left( \cos 3x - \cos x \right)}{x^3}\]
\[\lim_{x \to 0} \frac{\sin \left( 3 + x \right) - \sin \left( 3 - x \right)}{x}\]
\[\lim_{x \to 0} \frac{\sin 3x + 7x}{4x + \sin 2x}\]
\[\lim_{x \to \pi} \frac{\sin x}{\pi - x}\]
\[\lim_{x \to \frac{\pi}{3}} \frac{\sqrt{3} - \tan x}{\pi - 3x}\]
\[\lim_{x \to a} \frac{\cos \sqrt{x} - \cos \sqrt{a}}{x - a}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin \pi x}\]
\[\lim_{x \to 1} \left( 1 - x \right) \tan \left( \frac{\pi x}{2} \right)\]
\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{1 - \sqrt{2} \sin x}\]
\[\lim_{x \to \frac{\pi}{2}} \frac{\left( \frac{\pi}{2} - x \right) \sin x - 2 \cos x}{\left( \frac{\pi}{2} - x \right) + \cot x}\]
Evaluate the following limit:
\[\lim_{x \to \pi} \frac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left( \cos\frac{x}{4} - \sin\frac{x}{4} \right)}\]
\[\lim_{x \to 0} \frac{8^x - 2^x}{x}\]
\[\lim_{x \to \infty} \left\{ \frac{3 x^2 + 1}{4 x^2 - 1} \right\}^\frac{x^3}{1 + x}\]
\[\lim_{n \to \infty} \frac{1^2 + 2^2 + 3^2 + . . . + n^2}{n^3}\]
\[\lim_{x \to 0} \frac{\left| \sin x \right|}{x}\]
If the value of `lim_(x -> 1) (1 - (1 - x))^"m"/x` is 99, then n = ______.
Evaluate the following limit:
`lim_(x->3)[(sqrt(x+6))/x]`
Evaluate the Following limit:
`lim_(x->7)[((root(3)(x)-root(3)(7))(root(3)(x)+root(3)(7)))/(x-7)]`
