Advertisements
Advertisements
प्रश्न
Evaluate the following limit:
\[\lim_{x \to \pi} \frac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left( \cos\frac{x}{4} - \sin\frac{x}{4} \right)}\]
Advertisements
उत्तर
\[\lim_{x \to \pi} \frac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left( \cos\frac{x}{4} - \sin\frac{x}{4} \right)}\]
Put
\[x = \pi + h\] When \[x \to \pi, h \to 0\]
\[\therefore \lim_{x \to \pi} \frac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left( \cos\frac{x}{4} - \sin\frac{x}{4} \right)}\]
\[ = \lim_{h \to 0} \frac{1 - \sin\left( \frac{\pi + h}{2} \right)}{\cos\left( \frac{\pi + h}{2} \right)\left[ \cos\left( \frac{\pi + h}{4} \right) - \sin\left( \frac{\pi + h}{4} \right) \right]}\]
\[ = \lim_{h \to 0} \frac{1 - \sin\left( \frac{\pi}{2} + \frac{h}{2} \right)}{\cos\left( \frac{\pi}{2} + \frac{h}{2} \right)\left[ \cos\left( \frac{\pi}{4} + \frac{h}{4} \right) - \sin\left( \frac{\pi}{4} + \frac{h}{4} \right) \right]}\]
\[ = \lim_{h \to 0} \frac{1 - \cos\left( \frac{h}{2} \right)}{- \sin\left( \frac{h}{2} \right)\left[ \left( \cos\frac{\pi}{4}\cos\frac{h}{4} - \sin\frac{\pi}{4}\sin\frac{h}{4} \right) - \left( \sin\frac{\pi}{4}\cos\frac{h}{4} + \cos\frac{\pi}{4}\sin\frac{h}{4} \right) \right]}\]
\[= \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{4}}{- 2\sin\frac{h}{4}\cos\frac{h}{4}\left( \frac{1}{\sqrt{2}}\cos\frac{h}{4} - \frac{1}{\sqrt{2}}\sin\frac{h}{4} - \frac{1}{\sqrt{2}}\cos\frac{h}{4} - \frac{1}{\sqrt{2}}\sin\frac{h}{4} \right)}\]
\[ = \lim_{h \to 0} \frac{\sin\frac{h}{4}}{- \cos\frac{h}{4} \times \left( - \sqrt{2}\sin\frac{h}{4} \right)}\]
\[ = \frac{1}{\sqrt{2}} \times \frac{1}{\lim_{h \to 0} \cos\frac{h}{4}}\]
\[ = \frac{1}{\sqrt{2}} \times 1 \left( \lim_\theta \to 0 \cos\theta = 1 \right)\]
\[ = \frac{1}{\sqrt{2}}\]
APPEARS IN
संबंधित प्रश्न
\[\lim_{x \to 1} \frac{\sqrt{x + 8}}{\sqrt{x}}\]
\[\lim_{x \to - 1}{\left( 4 x^2 + 2 \right)}\]
\[\lim_{x \to - 1} \frac{x^3 - 3x + 1}{x - 1}\]
\[\lim_{x \to 0} \frac{\left( a + x \right)^2 - a^2}{x}\]
\[\lim_{x \to 3} \frac{x^2 - x - 6}{x^3 - 3 x^2 + x - 3}\]
\[\lim_{x \to a} \frac{\left( x + 2 \right)^{5/2} - \left( a + 2 \right)^{5/2}}{x - a}\]
\[\lim_{x \to a} \frac{x^{5/7} - a^{5/7}}{x^{2/7} - a^{2/7}}\]
\[\lim_{x \to 27} \frac{\left( x^{1/3} + 3 \right) \left( x^{1/3} - 3 \right)}{x - 27}\]
\[\lim_{x \to \infty} \frac{5 x^3 - 6}{\sqrt{9 + 4 x^6}}\]
\[\lim_{n \to \infty} \frac{n^2}{1 + 2 + 3 + . . . + n}\]
\[\lim_{x \to \infty} \frac{3 x^{- 1} + 4 x^{- 2}}{5 x^{- 1} + 6 x^{- 2}}\]
\[\lim_{x \to - \infty} \left( \sqrt{4 x^2 - 7x} + 2x \right)\]
\[\lim_{x \to 0} \frac{\sin 3x}{5x}\]
\[\lim_{x \to 0} \frac{\tan 3x - 2x}{3x - \sin^2 x}\]
\[\lim_{x \to 0} \frac{\sin \left( 2 + x \right) - \sin \left( 2 - x \right)}{x}\]
\[\lim_{x \to 0} \frac{x^2 + 1 - \cos x}{x \sin x}\]
Evaluate the following limits:
\[\lim_{x \to 0} \frac{\cos ax - \cos bx}{\cos cx - 1}\]
\[\lim_{x \to a} \frac{\cos x - \cos a}{x - a}\]
\[\lim_{x \to \frac{\pi}{2}} \frac{\sqrt{2 - \sin x} - 1}{\left( \frac{\pi}{2} - x \right)^2}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin 2\pi x}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin \pi x}\]
\[\lim_{x \to 0} \frac{8^x - 2^x}{x}\]
\[\lim_{x \to 0^+} \left\{ 1 + \tan^2 \sqrt{x} \right\}^{1/2x}\]
Write the value of \[\lim_{x \to 0^+} \left[ x \right] .\]
Write the value of \[\lim_{x \to 2} \frac{\left| x - 2 \right|}{x - 2} .\]
Write the value of \[\lim_{x \to 0} \frac{\sin x^\circ}{x} .\]
\[\lim_{n \to \infty} \frac{1^2 + 2^2 + 3^2 + . . . + n^2}{n^3}\]
\[\lim_{x \to 0} \frac{\sin 2x}{x}\]
\[\lim_{x \to 0} \frac{\left( 1 - \cos 2x \right) \sin 5x}{x^2 \sin 3x} =\]
\[\lim_{x \to 0} \frac{x}{\tan x} is\]
\[\lim_{h \to 0} \left\{ \frac{1}{h\sqrt[3]{8 + h}} - \frac{1}{2h} \right\} =\]
\[\lim_\theta \to \pi/2 \frac{1 - \sin \theta}{\left( \pi/2 - \theta \right) \cos \theta}\] is equal to
The value of \[\lim_{x \to \pi/2} \left( \sec x - \tan x \right)\]is
Evaluate the following limit:
`lim_(x -> 3) [sqrt(x + 6)/x]`
Evaluate the following limits: if `lim_(x -> 1)[(x^4 - 1)/(x - 1)] = lim_(x -> "a") [(x^3 - "a"^3)/(x - "a")]`, find all the value of a.
Evaluate the following limit:
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
Evaluate the following Limit:
`lim_(x -> 0) ((1 + x)^"n" - 1)/x`
Evaluate the following limits: `lim_(x ->3) [sqrt(x + 6)/x]`
Evaluate the following limit:
`lim_(x->3)[(sqrt(x+6))/x]`
