Advertisements
Advertisements
प्रश्न
\[\lim_{x \to \infty} \frac{\sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}}{\sqrt{x^2 + c^2} - \sqrt{x^2 + d^2}}\]
Advertisements
उत्तर
\[\lim_{x \to \infty} \left[ \frac{\sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}}{\sqrt{x^2 + c^2} - \sqrt{x^2 + d^2}} \right]\]
\[\text{ Rationalising the numerator and the denominator }:\]
\[ \lim_{x \to \infty} \left[ \frac{\left( \sqrt{x^2 + a^2} - \sqrt{x^2 + b^2} \right)}{\left( \sqrt{x^2 + c^2} - \sqrt{x^2 + d^2} \right)} \times \frac{\left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right)}{\left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right)} \times \frac{\left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right)}{\left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right)} \right]\]
\[ = \lim_{x \to \infty} \left[ \frac{\left( \sqrt{x^2 + a^2} - \sqrt{x^2 + b^2} \right) \left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right) \left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right)}{\left( \sqrt{x^2 + c^2} - \sqrt{x^2 + d^2} \right) \left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right) \left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right)} \right]\]
\[ = \lim_{x \to \infty} \frac{\left( x^2 + a^2 \right) - \left( x^2 + b^2 \right)}{\left( x^2 + c^2 \right) - \left( x^2 + d^2 \right)} \times \left( \frac{\sqrt{x^2 + c^2} + \sqrt{x^2 + d^2}}{\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2}} \right)\]
\[ {= \lim}_{x \to \infty} \left( \frac{a^2 - b^2}{c^2 - d^2} \right) \left( \frac{\sqrt{x^2 + c^2} + \sqrt{x^2 + d^2}}{\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2}} \right)\]
\[\text{ Dividing the numerator and the denominator byx }:\]
\[ \lim_{x \to \infty} \left( \frac{a^2 - b^2}{c^2 - d^2} \right) \left( \frac{\sqrt{1 + \frac{c^2}{x^2}} + \sqrt{1 + \frac{d^2}{x^2}}}{\sqrt{1 + \frac{1}{x^2}} + \sqrt{1 + \frac{b^2}{x^2}}} \right)\]
\[As x \to \infty , \frac{1}{x}, \frac{1}{x^2} \to 0\]
\[ = \left( \frac{a^2 - b^2}{c^2 - d^2} \right) \left( \frac{\sqrt{1} + \sqrt{1}}{\sqrt{1} + \sqrt{1}} \right)\]
\[ = \frac{a^2 - b^2}{c^2 - d^2}\]
APPEARS IN
संबंधित प्रश्न
\[\lim_{x \to 5} \frac{x^3 - 125}{x^2 - 7x + 10}\]
\[\lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}\]
\[\lim_{x \to \infty} \frac{3 x^3 - 4 x^2 + 6x - 1}{2 x^3 + x^2 - 5x + 7}\]
\[\lim_{x \to \infty} \frac{x}{\sqrt{4 x^2 + 1} - 1}\]
\[\lim_{x \to - \infty} \left( \sqrt{x^2 - 8x} + x \right)\]
\[\lim_{x \to 0} \frac{\tan^2 3x}{x^2}\]
\[\lim_\theta \to 0 \frac{\sin 3\theta}{\tan 2\theta}\]
\[\lim_{x \to 0} \frac{\sin x^2 \left( 1 - \cos x^2 \right)}{x^6}\]
\[\lim_{x \to 0} \frac{\sin 3x - \sin x}{\sin x}\]
\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos x}\]
\[\lim_{x \to 0} \frac{x^2 + 1 - \cos x}{x \sin x}\]
\[\lim_{x \to 0} \frac{5x + 4 \sin 3x}{4 \sin 2x + 7x}\]
If \[\lim_{x \to 0} kx cosec x = \lim_{x \to 0} x cosec kx,\]
\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{\cos x} - \sqrt{\sin x}}{x - \frac{\pi}{4}}\]
\[\lim_{x \to \frac{\pi}{2}} \frac{\sqrt{2 - \sin x} - 1}{\left( \frac{\pi}{2} - x \right)^2}\]
\[\lim_{x \to \pi} \frac{\sqrt{5 + \cos x} - 2}{\left( \pi - x \right)^2}\]
\[\lim_{x \to a} \frac{\sin \sqrt{x} - \sin \sqrt{a}}{x - a}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin 2\pi x}\]
\[\lim_{x \to \frac{\pi}{4}} \frac{f\left( x \right) - f\left( \frac{\pi}{4} \right)}{x - \frac{\pi}{4}},\]
\[\lim_{x \to 1} \frac{1 + \cos \pi x}{\left( 1 - x \right)^2}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin \pi x}\]
\[\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\sin \pi \left( x - 1 \right)}\]
\[\lim_{n \to \infty} 2^{n - 1} \sin \left( \frac{a}{2^n} \right)\]
\[\lim_{x \to \pi} \frac{1 + \cos x}{\tan^2 x}\]
\[\lim_{x \to \pi} \frac{\sqrt{2 + \cos x} - 1}{\left( \pi - x \right)^2}\]
\[\lim_{x \to 0} \frac{\log \left( a + x \right) - \log a}{x}\]
Write the value of \[\lim_{x \to 0^-} \left[ x \right] .\]
Write the value of \[\lim_{x \to 0} \frac{\sin x^\circ}{x} .\]
If \[f\left( x \right) = x \sin \left( 1/x \right), x \neq 0,\] then \[\lim_{x \to 0} f\left( x \right) =\]
\[\lim_{x \to \pi/4} \frac{\sqrt{2} \cos x - 1}{\cot x - 1}\] is equal to
\[\lim_{h \to 0} \left\{ \frac{1}{h\sqrt[3]{8 + h}} - \frac{1}{2h} \right\} =\]
\[\lim_{x \to \pi/3} \frac{\sin \left( \frac{\pi}{3} - x \right)}{2 \cos x - 1}\] is equal to
If α is a repeated root of ax2 + bx + c = 0, then \[\lim_{x \to \alpha} \frac{\tan \left( a x^2 + bx + c \right)}{\left( x - \alpha \right)^2}\]
Evaluate the following limits: `lim_(x -> 5)[(x^3 - 125)/(x^2 - 25)]`
Evaluate: `lim_(x -> 1) ((1 + x)^6 - 1)/((1 + x)^2 - 1)`
If `lim_(x -> 1) (x^4 - 1)/(x - 1) = lim_(x -> k) (x^3 - l^3)/(x^2 - k^2)`, then find the value of k.
Evaluate the following limits: `lim_(x ->3) [sqrt(x + 6)/x]`
Evaluate the following limit:
`lim_(x->3)[sqrt(x+6)/x]`
Evaluate the following limit:
`\underset{x->5}{lim}[(x^3 - 125)/(x^5 - 3125)]`
Evaluate the Following limit:
`lim_(x->7)[[(root[3][x] - root[3][7])(root[3][x] + root[3][7])] / (x - 7)]`
