If \( \alpha \) is the root of the equation \( x-\tan x=3 \) where \( \alpha \in\left(\frac{\pi}...
If \( \alpha \) is the root of the equation \( x-\tan x=3 \) where \( \alpha \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \); then which of the following is/are correct?, (where [.] denotes the greatest integer function and \( \{ \). fractional part function).
(a) \( \lim _{x \rightarrow \alpha^{+}}\left[\frac{\max (\tan x,\{x\})}{x-3}\right]=1 \)
(b) \( \lim _{x \rightarrow \alpha^{+}}\left[\frac{\min (\tan x,\{x\})}{x-3}\right]=1 \)
(c) \( \lim _{x \rightarrow \alpha^{-}}\left[\frac{\min (\tan x,\{x\})}{x-3}\right]=0 \)
(d) \( \lim _{x \rightarrow a^{-}}\left[\frac{\max (\tan x,\{x\})}{\tan x}\right]=1 \)
📲PW App Link - https://bit.ly/YTAI_PWAP
🌐PW Website - https://www.pw.live
Other Videos By PW Solutions
| 2023-06-08 | \( \lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^{4}} \) is equal to
(a) \( \frac{1}{3} ... |
| 2023-06-08 | The value of \( \lim _{x \rightarrow 1} \frac{\left(x^{2}-1\right) \sin ^{2}(\pi x)}{x^{4}-2 x^{... |
| 2023-06-08 | The value of \( \lim _{n \rightarrow \infty} \frac{[r]+[2 r]+\ldots+[n r]}{n^{2}} \), where \( r... |
| 2023-06-08 | If \( \lim _{x \rightarrow 0} \frac{a e^{x}-b \cos x+c e^{-x}}{x \sin x}=2 \), then \( a+b+c \) ... |
| 2023-06-08 | If \( \lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{3 x^{3}} \) is equal to \( l \), ... |
| 2023-06-08 | The value of \( \lim _{x \rightarrow 0^{+}} \frac{\cos ^{-1}\left(x-[x]^{2}\right) \cdot \sin ^{... |
| 2023-06-08 | If \( k=\lim _{x \rightarrow 1} \sec ^{-1}\left(\frac{\lambda^{2}}{\ln x}-\frac{\lambda^{2}}{x-1... |
| 2023-06-08 | Below is the graph of the function \( f \) :
Compute the following limits
(i) \( \lim _{x \right... |
| 2023-06-08 | Let \( x_{1}, x_{2}, x_{3} \) are roots of equation \( (x-1)(x-8)(x-31)=1 \). Then find the valu... |
| 2023-06-08 | If \( f(x)=\left\{\begin{array}{ll}\ln \operatorname{cosec}(x \pi) & 0x1 \\ \ln \sin (2 x \pi) &... |
| 2023-06-08 | If \( \alpha \) is the root of the equation \( x-\tan x=3 \) where \( \alpha \in\left(\frac{\pi}... |
| 2023-06-08 | Let \( f(x)=\max \{p, q, r\} \), where
\( p=\lim _{n \rightarrow \infty} \lim _{\alpha \rightarr... |
| 2023-06-08 | Let \( f(x)=\max \{p, q, r\} \), where
\( p=\lim _{n \rightarrow \infty} \lim _{\alpha \rightarr... |
| 2023-06-08 | Comprehension \( \quad: \) If the sequence is defined by \( a_{1}= \) 0 and \( a_{n+1}=a_{n}+4 n... |
| 2023-06-08 | Let \( f(x)=\max \{p, q, r\} \), where
\[
p=\lim _{n \rightarrow \infty} \lim _{\alpha \rightarr... |
| 2023-06-08 | Let \( f \) be a polynomial function satisfying \( f\left(x^{2}\right)-x f(x) \) \( =x^{4}\left(... |
| 2023-06-08 | \[
\lim _{x \rightarrow 0}\left(\frac{(1+x)^{\frac{2}{x}}}{e^{2}}\right)^{\frac{4}{\sin x}}
\]
(... |
| 2023-06-08 | \[
\lim _{x \rightarrow a} \frac{1}{\left(a^{2}-x^{2}\right)^{2}}\left[\frac{a^{2}+x^{2}}{a x}+\... |
| 2023-06-08 | If \( \alpha, \beta \) are two distinct real roots of the equation \( a x^{3}+x-1-a \) \( =0 \),... |
| 2023-06-08 | \[
\lim _{x \rightarrow \infty}\left(\frac{(x+1)^{x}}{x^{x} \cdot e}\right)^{x}=
\]
(a) \( e^{1 ... |
| 2023-06-08 | \( \lim _{x \rightarrow 0} \frac{1-\frac{4}{\frac{1}{f(x)}+\frac{1}{f(2 x)}+\frac{1}{f(3 x)}+\fr... |
0