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
Suppose f(x) = `{(a+bx, x < 1),(4, x = 1),(b-ax, x > 1):}` and if `lim_(x -> 1) f(x) = f(1)` what are possible values of a and b?
\[\lim_{x \to 1} \frac{\sqrt{x + 8}}{\sqrt{x}}\]
\[\lim_{x \to 4} \frac{x^2 - 7x + 12}{x^2 - 3x - 4}\]
\[\lim_{x \to 1} \frac{x^4 - 3 x^3 + 2}{x^3 - 5 x^2 + 3x + 1}\]
\[\lim_{x \to 1} \frac{x^3 + 3 x^2 - 6x + 2}{x^3 + 3 x^2 - 3x - 1}\]
Evaluate the following limit:
\[\lim_{x \to 1} \frac{x^7 - 2 x^5 + 1}{x^3 - 3 x^2 + 2}\]
\[\lim_{x \to a} \frac{\left( x + 2 \right)^{3/2} - \left( a + 2 \right)^{3/2}}{x - a}\]
\[\lim_{x \to a} \frac{x^{2/7} - a^{2/7}}{x - a}\]
\[\lim_{x \to - 1/2} \frac{8 x^3 + 1}{2x + 1}\]
\[\lim_{x \to 27} \frac{\left( x^{1/3} + 3 \right) \left( x^{1/3} - 3 \right)}{x - 27}\]
If \[\lim_{x \to a} \frac{x^3 - a^3}{x - a} = \lim_{x \to 1} \frac{x^4 - 1}{x - 1},\] find all possible values of a.
\[\lim_{x \to \infty} \sqrt{x^2 + cx - x}\]
\[\lim_{n \to \infty} \frac{n^2}{1 + 2 + 3 + . . . + n}\]
\[\lim_{x \to \infty} \frac{3 x^{- 1} + 4 x^{- 2}}{5 x^{- 1} + 6 x^{- 2}}\]
\[\lim_{n \to \infty} \left[ \frac{\left( n + 2 \right)! + \left( n + 1 \right)!}{\left( n + 2 \right)! - \left( n + 1 \right)!} \right]\]
\[\lim_{x \to 0} \frac{x \cos x + 2 \sin x}{x^2 + \tan x}\]
\[\lim_{x \to 0} \frac{\tan 2x - \sin 2x}{x^3}\]
\[\lim_{x \to 0} \left( cosec x - \cot x \right)\]
Evaluate the following limit:
\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]
\[\lim_{x \to a} \frac{\cos \sqrt{x} - \cos \sqrt{a}}{x - a}\]
\[\lim_{x \to \pi} \frac{\sqrt{2 + \cos x} - 1}{\left( \pi - x \right)^2}\]
\[\lim_{x \to \frac{\pi}{6}} \frac{\cot^2 x - 3}{cosec x - 2}\]
\[\lim_{x \to \frac{3\pi}{2}} \frac{1 + {cosec}^3 x}{\cot^2 x}\]
\[\lim_{x \to 0} \frac{8^x - 2^x}{x}\]
\[\lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n\]
Write the value of \[\lim_{x \to 0^-} \left[ x \right] .\]
Write the value of \[\lim_{x \to 0^-} \frac{\sin \left[ x \right]}{\left[ x \right]} .\]
\[\lim_{x \to 3} \frac{x - 3}{\left| x - 3 \right|},\] is equal to
\[\lim_{x \to \infty} \frac{\sqrt{x^2 - 1}}{2x + 1}\]
\[\lim_{x \to \infty} a^x \sin \left( \frac{b}{a^x} \right), a, b > 1\] is equal to
The value of \[\lim_{x \to 0} \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}\]
The value of \[\lim_{x \to \infty} \frac{n!}{\left( n + 1 \right)! - n!}\]
If \[f\left( x \right) = \begin{cases}\frac{\sin\left[ x \right]}{\left[ x \right]}, & \left[ x \right] \neq 0 \\ 0, & \left[ x \right] = 0\end{cases}\] where denotes the greatest integer function, then \[\lim_{x \to 0} f\left( x \right)\]
Evaluate the following limits: `lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`
Evaluate the following limits: if `lim_(x -> 5)[(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
Evaluate the following limit :
`lim_(x->5)[(x^3-125)/(x^5-3125)]`
Evaluate the following limit:
`lim_(x->3)[sqrt(x+6)/x]`
Evaluate the following limit:
`lim_(x->5)[(x^3-125)/(x^5-3125)]`
Evaluate the following limit:
`lim _ (x -> 5) [(x^3 - 125) / (x^5 - 3125)]`
