Advertisements
Advertisements
प्रश्न
Write the value of \[\lim_{x \to \pi/2} \frac{2x - \pi}{\cos x} .\]
Advertisements
उत्तर
\[\lim_{x \to \frac{\pi}{2}} \left[ \frac{2x - \pi}{\cos x} \right]\]
\[LHL: \]
\[ \lim_{x \to \frac{\pi}{2}^-} \left[ \frac{2x - \pi}{\cos x} \right]\]
\[Let x = \frac{\pi}{2} - h\]
\[\text{ If } x \to \frac{\pi}{2}, \text{ then we have }: \]
\[ h \to 0\]
\[ = \lim_{h \to 0} \left[ \frac{2\left( \frac{\pi}{2} - h \right) - \pi}{\cos \left( \frac{\pi}{2} - h \right)} \right]\]
\[ = \lim_{h \to 0} \left[ \frac{\pi - 2h - \pi}{\sin h} \right]\]
\[ = - 2\]
\[RHL: \]
\[ \lim_{x \to \frac{\pi}{2}^+} \left[ \frac{2x - \pi}{\cos x} \right]\]
\[\text{ Let } x = \frac{\pi}{2} + h\]
\[\text{ If } x \to \frac{\pi}{2}, \text{ then we have }: \]
\[h \to 0\]
\[ = \lim_{h \to 0} \left[ \frac{2\left( \frac{\pi}{2} + h \right) - \pi}{\cos \left( \frac{\pi}{2} + h \right)} \right]\]
\[ = \lim_{h \to 0} \left[ \frac{2h}{- \sin h} \right]\]
\[ = - 2\]
\[ \Rightarrow \lim_{x \to \frac{\pi}{2}} \left( \frac{2x - \pi}{\cos x} \right) = - 2\]
APPEARS IN
संबंधित प्रश्न
Let a1, a2,..., an be fixed real numbers and define a function f ( x) = ( x − a1 ) ( x − a2 )...( x − an ).
What is `lim_(x -> a_1) f(x)` ? For some a ≠ a1, a2, ..., an, compute `lim_(x -> a) f(x)`
If f(x) = `{(|x| + 1,x < 0), (0, x = 0),(|x| -1, x > 0):}`
For what value (s) of a does `lim_(x -> a)` f(x) exists?
if `f(x) = { (mx^2 + n, x < 0),(nx + m, 0<= x <= 1),(nx^3 + m, x > 1):}`
For what integers m and n does `lim_(x-> 0) f(x)` and `lim_(x -> 1) f(x)` exist?
\[\lim_{x \to 2} \frac{\sqrt{3 - x} - 1}{2 - x}\]
\[\lim_{x \to 1} \frac{\sqrt{5x - 4} - \sqrt{x}}{x - 1}\]
\[\lim_{x \to 2} \frac{\sqrt{1 + 4x} - \sqrt{5 + 2x}}{x - 2}\]
\[\lim_{x \to 0} \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{x}\]
\[\lim_{x \to 0} \frac{\sqrt{2 - x} - \sqrt{2 + x}}{x}\]
\[\lim_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^2 - 1}\]
\[\lim_{x \to 1} \frac{ x^2 - \sqrt{x}}{\sqrt{x} - 1}\]
\[\lim_{x \to \sqrt{2}} \frac{\sqrt{3 + 2x} - \left( \sqrt{2} + 1 \right)}{x^2 - 2}\]
\[\lim_{x \to 0} \frac{a^x + a^{- x} - 2}{x^2}\]
\[\lim_{x \to 0} \frac{a^x + b^x - 2}{x}\]
\[\lim_{x \to 0} \frac{9^x - 2 . 6^x + 4^x}{x^2}\]
\[\lim_{x \to 2} \frac{x - 2}{\log_a \left( x - 1 \right)}\]
\[\lim_{x \to 0} \frac{5^x + 3^x + 2^x - 3}{x}\]
\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{\sin kx}\]
\[\lim_{x \to 0} \frac{\sin 2x}{e^x - 1}\]
\[\lim_{x \to a} \frac{\log x - \log a}{x - a}\]
\[\lim_{x \to 0} \frac{\log \left( 2 + x \right) + \log 0 . 5}{x}\]
\[\lim_{x \to 0} \frac{x\left( 2^x - 1 \right)}{1 - \cos x}\]
\[\lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}}\]
\[\lim_{x \to 0} \frac{e^{3 + x} - \sin x - e^3}{x}\]
`\lim_{x \to 0} \frac{e^\tan x - 1}{\tan x}`
\[\lim_{x \to 0} \frac{e^{bx} - e^{ax}}{x} \text{ where } 0 < a < b\]
\[\lim_{x \to 0} \frac{3^{2 + x} - 9}{x}\]
\[\lim_{x \to 0} \frac{a^x - a^{- x}}{x}\]
\[\lim_{x \to \infty} \left\{ \frac{x^2 + 2x + 3}{2 x^2 + x + 5} \right\}^\frac{3x - 2}{3x + 2}\]
\[\lim_{x \to 0} \frac{\sin x}{\sqrt{1 + x} - 1} .\]
Write the value of \[\lim_{x \to - \infty} \left( 3x + \sqrt{9 x^2 - x} \right) .\]
Write the value of \[\lim_{n \to \infty} \frac{n! + \left( n + 1 \right)!}{\left( n + 1 \right)! + \left( n + 2 \right)!} .\]
