Advertisements
Advertisements
प्रश्न
For what value of k is the function
\[f\left( x \right) = \begin{cases}\frac{\sin 5x}{3x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0?\]
उत्तर
Given:
\[f\left( x \right) = \binom{\frac{\sin5x}{3x}, if x \neq 0}{k, if x = 0}\]
If
\[f\left( x \right)\] is continuous at x = 0, then
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[\lim_{x \to 0} \frac{\sin5x}{3x} = k\]
\[\lim_{x \to 0} \frac{5 \sin5x}{3 \times 5x} = k\]
\[\frac{5}{3} \lim_{x \to 0} \frac{\sin5x}{5x} = k\]
\[\frac{5}{3} \times 1 = k\]
\[k = \frac{5}{3}\]
संबंधित प्रश्न
Examine the continuity of the following function :
`{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
Examine the following function for continuity:
f (x) = x – 5
Examine the following function for continuity:
`f (x)1/(x - 5), x != 5`
If \[f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}\] find whether f is continuous at x = 0.
Discuss the continuity of \[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0\]
For what value of k is the following function continuous at x = 1? \[f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k , & x = 1\end{cases}\]
Determine the values of a, b, c for which the function f(x) = `{((sin(a + 1)x + sin x)/x, "for" x < 0),(x, "for" x = 0),((sqrt(x + bx^2) - sqrtx)/(bx^(3"/"2)), "for" x > 0):}` is continuous at x = 0.
If \[f\left( x \right) = \begin{cases}\frac{x - 4}{\left| x - 4 \right|} + a, \text{ if } & x < 4 \\ a + b , \text{ if } & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, \text{ if } & x > 4\end{cases}\] is continuous at x = 4, find a, b.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, & \text{ if } x \neq 0 \\ 4 , & \text{ if } x = 0\end{cases}\]
Find the points of discontinuity, if any, of the following functions:
Write the value of b for which \[f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}\] is continuous at x = 1.
If the function \[f\left( x \right) = \begin{cases}\left( \cos x \right)^{1/x} , & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then the value of k is
If \[f\left( x \right) = \begin{cases}mx + 1 , & x \leq \frac{\pi}{2} \\ \sin x + n, & x > \frac{\pi}{2}\end{cases}\] is continuous at \[x = \frac{\pi}{2}\] , then
The value of f (0), so that the function
If \[f\left( x \right) = \frac{1}{1 - x}\] , then the set of points discontinuity of the function f (f(f(x))) is
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Show that f(x) = x1/3 is not differentiable at x = 0.
Show that the function
(i) differentiable at x = 0, if m > 1
(ii) continuous but not differentiable at x = 0, if 0 < m < 1
(iii) neither continuous nor differentiable, if m ≤ 0
Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]
The function f (x) = e−|x| is
If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]
The function f (x) = |cos x| is
The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is
Examine the continuity of f(x)=`x^2-x+9 "for" x<=3`
=`4x+3 "for" x>3, "at" x=3`
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
`f(x)=(x^2-9)/(x - 3)` is not defined at x = 3. what value should be assigned to f(3) for continuity of f(x) at = 3?
If f is continuous at x = 0 then find f(0) where f(x) = `[5^x + 5^-x - 2]/x^2`, x ≠ 0
If the function f is continuous at x = 2, then find 'k' where
f(x) = `(x^2 + 5)/(x - 1),` for 1< x ≤ 2
= kx + 1 , for x > 2
Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`
If f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.
Let f(x) = `{{:((1 - cos 4x)/x^2",", "if" x < 0),("a"",", "if" x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if" x > 0):}`. For what value of a, f is continuous at x = 0?
Examine the differentiability of the function f defined by
f(x) = `{{:(2x + 3",", "if" -3 ≤ x < - 2),(x + 1",", "if" -2 ≤ x < 0),(x + 2",", "if" 0 ≤ x ≤ 1):}`
The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.
For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).
Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1
f(x) = `{{:(|x - "a"| sin 1/(x - "a")",", "if" x ≠ 0),(0",", "if" x = "a"):}` at x = a
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.
