Advertisements
Advertisements
प्रश्न
Find the values 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`
उत्तर १
The given function f is `f(x) = {((kcosx)/(pi-2x), "," if x != pi/2),(3, "," if x = pi/2):} " at x " pi/2`
उत्तर २
`f(x) = {(("k cos x")/(pi - 2"x")"," " if" "x" ne pi/2),(3"," " if" "x" = pi/2):}`
If f(x) is continuous at x = `pi/2`, it implies:
L.H.L. = `lim_(x -> pi^-/2) f(x) = lim_(h -> 0) (pi/2 - h)`
`= lim_(h -> 0) (k cos (pi/2 - h))/(pi - 2(pi/2 - h))`
`= lim_(h -> 0) (k sin h)/(pi - pi + 2h)`
`= lim_(h -> 0) k/2 (sin h)/h = k/2 ...(because lim_("h" -> 0) (sin h)/h = 1)`
R.H.L. = `lim_(x -> pi^+/2) f(x) = lim_(h -> 0) (pi/2 + h)`
`= lim_(h -> 0) (k cos (pi/2 + h))/(pi - 2(pi/2 + h))`
`= lim_(h -> 0) (- k sin h)/(- 2h)`
`= lim_(h -> 0) k/2 (sin h)/h = k/2`
`f(pi/2) = 3`
The function f will be continuous at x = `pi/2` if
L.H.L. = R.H.L. = `f(pi/2)`
`therefore k/2 = 3`
⇒ k = 6
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`
Is the function defined by `f(x) = x^2 - sin x + 5` continuous at x = π?
Show that the function defined by f(x) = |cos x| is a continuous function.
Examine sin |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 \[f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x}, & x \neq \pi/2 \\ 3 , & x = \pi/2\end{cases}\]
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
Find the values of a and b so that the function f given by \[f\left( x \right) = \begin{cases}1 , & \text{ if } x \leq 3 \\ ax + b , & \text{ if } 3 < x < 5 \\ 7 , & \text{ if } x \geq 5\end{cases}\] is continuous at x = 3 and x = 5.
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.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, & \text{ if } x \neq 0 \\ 3k , & \text{ if } x = 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if } x < 0 \\ \cos 2x , & \text{ if } x \geq 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}4 , & \text{ if } x \leq - 1 \\ a x^2 + b, & \text{ if } - 1 < x < 0 \\ \cos x, &\text{ if }x \geq 0\end{cases}\]
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
Discuss the continuity of f(x) = sin | x |.
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.
Determine the value of the constant 'k' so that function f
If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to
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
The function
Find the values of a and b so that the function
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
Let f(x) = |sin x|. Then ______.
The function f(x) = x2 – sin x + 5 is continuous at x =
What is the values of' 'k' so that the function 'f' is continuous at the indicated point
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx + 1",", if x ≤ pi),(cos x",", if x > pi):}` at = `pi`
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.
Discuss the continuity of the following function:
f(x) = sin x – cos x
