The value of k for which the function f(x) = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For the function to be continuous at $x = \frac{\pi}{2}$,we must have $\lim_{x 	o \frac{\pi}{2}} f(x) = f(\frac{\pi}{2})$.First,calculate the lim

Vedclass · Accepted Answer

$\frac{3}{5}$

Vedclass · Answer

(C) For the function to be continuous at $x = \frac{\pi}{2}$,we must have $\lim_{x 	o \frac{\pi}{2}} f(x) = f(\frac{\pi}{2})$.First,calculate the limit: $\lim_{x 	o \frac{\pi}{2}} (\frac{4}{5})^{\frac{	an 4x}{	an 5x}}$.Let $x = \frac{\pi}{2} + h$,where $h 	o 0$.Then $	an 4x = 	an(4(\frac{\pi}{2} + h)) = 	an(2\pi + 4h) = 	an 4h \approx 4h$.And $	an 5x = 	an(5(\frac{\pi}{2} + h)) = 	an(\frac{5\pi}{2} + 5h) = \cot 5h = \frac{1}{	an 5h} \approx \frac{1}{5h}$.Thus,$\frac{	an 4x}{	an 5x} = \frac{	an 4h}{\cot 5h} = 	an 4h \cdot 	an 5h 	o 0 \cdot \infty$ (indeterminate form).Actually,$\frac{	an 4x}{	an 5x} = \frac{	an(2\pi + 4h)}{	an(\frac{5\pi}{2} + 5h)} = \frac{	an 4h}{\cot 5h} = 	an 4h \cdot 	an 5h$. As $h 	o 0$,this limit is $0 \cdot 0 = 0$.So,$\lim_{x 	o \frac{\pi}{2}} f(x) = (\frac{4}{5})^0 = 1$.Equating to $f(\frac{\pi}{2}) = k + \frac{2}{5}$,we get $k + \frac{2}{5} = 1$.Therefore,$k = 1 - \frac{2}{5} = \frac{3}{5}$.

The value of $k$ for which the function $f(x) = \begin{cases} (\frac{4}{5})^{\frac{\tan 4x}{\tan 5x}}, & 0 < x < \frac{\pi}{2} \\ k + \frac{2}{5}, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$ is

Similar Questions

$6 \Omega$ and $12 \Omega$ resistors are connected in parallel. This combination is connected in series with a $10 \text{ V}$ battery and a $6 \Omega$ resistor. What is the potential difference between the terminals of the $12 \Omega$ resistor (in $\text{ V}$)?

Thorn is a modified branch because:

If $f(x) = \left|\begin{array}{ccc} 1 & x & x+1 \\ 2x & x(x-1) & x(x+1) \\ 3x(x-1) & x(x-1)(x-2) & (x-1)x(x+1) \end{array}\right|$,then $f(2012)$ is equal to:

If $1$ is a multiple root of order $3$ for the equation $x^4-2x^3+2x-1=0$,then the other root is

The equivalent resistance of a series combination of two resistors is $s$. When they are connected in parallel,the equivalent resistance is $p$. If $s = np$,then the minimum value for $n$ is (Round off to the nearest integer).

Vedclass Test Series

Exam Paper Generator

Online Exam Module