Advertisements
Advertisements
प्रश्न
If \[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, write the value of k.
Advertisements
उत्तर
Given,
\[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]
If \[f\left( x \right)\] is continuous at \[x = 0\] , then
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[\Rightarrow \lim_{x \to 0} \left( \frac{\sin^{- 1} x}{x} \right) = k\]
\[ \Rightarrow k = 1 \left[ \because \lim_{x \to 0} \left( \frac{\sin^{- 1} x}{x} \right) = 1 \right]\]
APPEARS IN
संबंधित प्रश्न
For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if" x <= 0),(4x+ 1", if" x > 0):}` continuous at x = 0? What about continuity at x = 1?
Is the function defined by f(x) = x2 − sin x + 5 continuous at x = π?
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 value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx +1", if" x<= pi),(cos x", if" x > pi):}` at x = π
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 x| is a continuous function.
Determine the value of the constant k so that the function
\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0 .\]
Let \[f\left( x \right) = \frac{\log\left( 1 + \frac{x}{a} \right) - \log\left( 1 - \frac{x}{b} \right)}{x}\] x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.
If \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin } x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (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.
If \[f\left( x \right) = \begin{cases}2 x^2 + k, &\text{ if } x \geq 0 \\ - 2 x^2 + k, & \text{ if } x < 0\end{cases}\] then what should be the value of k so that f(x) is continuous at x = 0.
Discuss the continuity of the function
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & \text{ if } x < 0 \\ 2x + 3, & x \geq 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, & \text{ if } x \neq 0 \\ 3k , & \text{ if } x = 0\end{cases}\]
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.
Show that f (x) = | cos x | is a continuous function.
What happens to a function f (x) at x = a, if
Determine whether \[f\left( x \right) = \binom{\frac{\sin x^2}{x}, x \neq 0}{0, x = 0}\] is continuous at x = 0 or not.
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
The function
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
Find the values of a and b so that the function
If f is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
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
Let f(x) = |sin x|. Then ______.
If f.g is continuous at x = a, then f and g are separately continuous at x = a.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
A real value of x satisfies `((3 - 4ix)/(3 + 4ix))` = α – iβ (α, β ∈ R), if α2 + β2 is equal to
The function f(x) = 5x – 3 is continuous at x =
Find the values of `a` and ` b` such that the function by:
`f(x) = {{:(5",", if x ≤ 2),(ax + b",", if 2 < x < 10),(21",", if x ≥ 10):}`
is a continuous function.
The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",", if x ≠ 0),(k",", if x = 0):}` is continuous at x = 0 is ______.
Discuss the continuity of the following function:
f(x) = sin x + cos x
