Question:
The $\displaystyle \lim_{x \to \frac{\pi}{2}} \left\{ 2x \tan x - \frac{\pi}{\cos x }\right\} $ is
The $\displaystyle \lim_{x \to \frac{\pi}{2}} \left\{ 2x \tan x - \frac{\pi}{\cos x }\right\} $ is
Updated On: Jun 20, 2022
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The Correct Option is C
Solution and Explanation
Let $L =\displaystyle \lim _{x \rightarrow \frac{\pi}{2}}\left\{2 x \tan x-\frac{\pi}{\cos X}\right\}$
$=\displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left\{\frac{2 x \sin x-\pi}{\cos \,x}\right\} $
$=\displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left\{2 x \frac{\sin x}{\cos X}-\frac{\pi}{\cos x}\right\} $
$[\frac{0}{0} $ form ]
Now, by using L'Hospital Rule
$L= \displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left\{\frac{2 \sin x+2 x \cos x}{(-\sin x)}\right\} $
$= \frac{2 \times 1+2 \times \frac{\pi}{2} \times \cos \frac{\pi}{2}}{(-1)}=-2$
$=\displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left\{\frac{2 x \sin x-\pi}{\cos \,x}\right\} $
$=\displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left\{2 x \frac{\sin x}{\cos X}-\frac{\pi}{\cos x}\right\} $
$[\frac{0}{0} $ form ]
Now, by using L'Hospital Rule
$L= \displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left\{\frac{2 \sin x+2 x \cos x}{(-\sin x)}\right\} $
$= \frac{2 \times 1+2 \times \frac{\pi}{2} \times \cos \frac{\pi}{2}}{(-1)}=-2$
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).