Advertisements
Advertisements
प्रश्न
If the following function is continuous at x = 2 then the value of k will be ______.
f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`
विकल्प
2
3
5
– 1
उत्तर
If the following function is continuous at x = 2 then the value of k will be 5.
f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`
Explanation:
k = 5
∵ f(x) is continuous at x = 2
∴ LHL = RHL = f(2) ...(i)
Now LHL = `lim_(h -> 0) f(2 - h)`
= `lim_(h -> 0) 2(2 - h) + 1`
= `lim_(h -> 0) 5 - 2h`
= 5 – 0
= 5
RHL = `lim_(h -> 0) f(2 + h)`
= `lim_(h -> 0) 3(2 + h) - 1`
= `lim_(h -> 0) 5 + 3h`
= 5 + 0
= 5
And f(2) = k
Then by (i),
LHL = RHL = f(2)
`\implies` k = 5
APPEARS IN
संबंधित प्रश्न
Examine the following function for continuity:
f(x) = | x – 5|
Discuss the continuity of the function f, where f is defined by `f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
If \[f\left( x \right) = \begin{cases}\frac{2^{x + 2} - 16}{4^x - 16}, \text{ if } & x \neq 2 \\ k , \text{ if } & x = 2\end{cases}\] is continuous at x = 2, find k.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}\]at x = 5
Prove that \[f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{x}, & x \neq 0 \\ 2 , & x = 0\end{cases}\] is discontinuous at x = 0
Define continuity of a function at a point.
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) = x1/3 is not differentiable at x = 0.
Is every differentiable function continuous?
Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]
Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4
= 10 for x = 4 at x = 4
Find the value of 'k' if the function
f(x) = `(tan 7x)/(2x)`, for x ≠ 0.
= k for x = 0.
is continuous at x = 0.
The probability distribution function of continuous random variable X is given by
f( x ) = `x/4`, 0 < x < 2
= 0, Otherwise
Find P( x ≤ 1)
Discuss the continuity of the function at the point given. If the function is discontinuous, then remove the discontinuity.
f (x) = `(sin^2 5x)/x^2` for x ≠ 0
= 5 for x = 0, at x = 0
If the function f is continuous at x = I, then find f(1), where f(x) = `(x^2 - 3x + 2)/(x - 1),` for x ≠ 1
If Y = tan-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`
The set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.
f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
Examine the differentiability of f, where f is defined by
f(x) = `{{:(x[x]",", "if" 0 ≤ x < 2),((x - 1)x",", "if" 2 ≤ x < 3):}` at x = 2
Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",", "if" x ≤ 1),("q"x + 2",", "if" x > 1):}` is differentiable at x = 1
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.
If f is continuous on its domain D, then |f| is also continuous on D.
