Advertisements
Advertisements
प्रश्न
Let \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .\] The value which should be assigned to f (x) at \[x = \frac{\pi}{4},\]so that it is continuous everywhere is
पर्याय
1
1/2
2
none of these
Advertisements
उत्तर
\[\frac{1}{2}\]
If \[f\left( x \right)\] is continuous at \[x = \frac{\pi}{4}\]
If \[\frac{\pi}{4} - x = y\], then
\[x \to \frac{\pi}{4} \text{ and } y \to 0\]
\[\therefore \lim_{y \to 0} \left( \frac{\tan y}{\cot 2\left( \frac{\pi}{4} - y \right)} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\tan y}{\cot\left( \frac{\pi}{2} - 2y \right)} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\tan y}{\tan 2y} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\frac{\tan y}{y}}{\frac{\tan 2y}{y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\frac{\tan y}{y}}{\frac{2 \tan 2y}{2y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \frac{1}{2} \lim_{y \to 0} \left( \frac{\frac{\tan y}{y}}{\frac{\tan 2y}{2y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \frac{1}{2}\left( \frac{\lim_{y \to 0} \frac{\tan y}{y}}{\lim_{y \to 0} \frac{\tan 2y}{2y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \frac{1}{2}\left( \frac{1}{1} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow f\left( \frac{\pi}{4} \right) = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
If f (x) is continuous on [–4, 2] defined as
f (x) = 6b – 3ax, for -4 ≤ x < –2
= 4x + 1, for –2 ≤ x ≤ 2
Show that a + b =`-7/6`
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{((kcosx)/(pi-2x)", if" x != pi/2),(3", if" x = pi/2):}` at x = `"pi/2`
Find the values of a and b such that the function defined by f(x) = `{(5", if" x <= 2),(ax +b", if" 2 < x < 10),(21", if" x >= 10):}` is a continuous function.
Show that the function defined by f(x) = cos (x2) is a continuous function.
Show that the function defined by f(x) = |cos x| is a continuous function.
Find the values of a so that the function
Find the value of k if f(x) is continuous at x = π/2, where \[f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x}, & x \neq \pi/2 \\ 3 , & x = \pi/2\end{cases}\]
Extend the definition of the following by continuity
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2kx}{x^2}, \text{ if } & x \neq 0 \\ 8 , \text{ if } & x = 0\end{cases}\] at x = 0
If \[f\left( x \right) = \begin{cases}\frac{x^2}{2}, & \text{ if } 0 \leq x \leq 1 \\ 2 x^2 - 3x + \frac{3}{2}, & \text P{ \text{ if } } 1 < x \leq 2\end{cases}\]. Show that f is continuous at x = 1.
Discuss the continuity of the function
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & \text{ if } x \neq 2 \\ 16 , & \text{ if } x = 2\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
Show that f (x) = cos x2 is a continuous function.
What happens to a function f (x) at x = a, if
If \[f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then write the value of k.
If \[f\left( x \right) = \begin{cases}\frac{\log\left( 1 + ax \right) - \log\left( 1 - bx \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] and f (x) is continuous at x = 0, then the value of k is
Let \[f\left( x \right) = \left\{ \begin{array}\\ \frac{x - 4}{\left| x - 4 \right|} + a, & x < 4 \\ a + b , & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, & x > 4\end{array} . \right.\]Then, f (x) is continuous at x = 4 when
\[f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)\] is continuous, is given by
The function
If \[f\left( x \right) = \frac{1 - \sin x}{\left( \pi - 2x \right)^2},\] when x ≠ π/2 and f (π/2) = λ, then f (x) will be continuous function at x= π/2, where λ =
The value of a for which the function \[f\left( x \right) = \begin{cases}\frac{\left( 4^x - 1 \right)^3}{\sin\left( x/a \right) \log \left\{ \left( 1 + x^2 /3 \right) \right\}}, & x \neq 0 \\ 12 \left( \log 4 \right)^3 , & x = 0\end{cases}\]may be continuous at x = 0 is
If the function f (x) defined by \[f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then k =
Find the values of a and b so that the function
The function f (x) = |cos x| is
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\]
then at x = 0, f (x) is
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
The function f(x) = `"e"^|x|` is ______.
Let f(x) = |sin x|. Then ______.
`lim_("x" -> 0) ("x cos x" - "log" (1 + "x"))/"x"^2` is equal to ____________.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
The value of f(0) for the function `f(x) = 1/x[log(1 + x) - log(1 - x)]` to be continuous at x = 0 should be
The function f(x) = x2 – sin x + 5 is continuous at x =
Discuss the continuity of the following function:
f(x) = sin x + cos x
