Advertisements
Advertisements
प्रश्न
f(x) = `{{:(3x - 8",", "if" x ≤ 5),(2"k"",", "if" x > 5):}` at x = 5
Advertisements
उत्तर
We have, f(x) = `{{:(3x - 8",", "if" x ≤ 5),(2"k"",", "if" x > 5):}`
Since, f(x) is continuous at x = 5
∴ L.H.L. = R.H.L. = f(5)
Now, L.H.L. = `lim_(x -> 5^-) (3x - 8)`
= `lim_("h" -> 0) [3(5 - "h") - 8]`
= –15 – 8
= 7
R.H.L. = `lim_(x -> 5^+) 2"k"`
= `lim_("h" -> 0) 2"k"` = 2k
Also f(5) = 3(5) – 8 = 7
∴ 2k = 7
⇒ k = `7/2`
APPEARS IN
संबंधित प्रश्न
If 'f' is continuous at x = 0, then find f(0).
`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`
Examine the following function for continuity:
f(x) = x – 5
Examine the following function for continuity:
f(x) = |x – 5|
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(3", if" 0 <= x <= 1),(4", if" 1 < x < 3),(5", if" 3 <= x <= 10):}`
A function f(x) is defined as
Show that f(x) is continuous at x = 3
Let \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x^2}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\] Show that f(x) is discontinuous at x = 0.
Discuss the continuity of the following functions at the indicated point(s):
Discuss the continuity of the function f(x) at the point x = 0, where \[f\left( x \right) = \begin{cases}x, x > 0 \\ 1, x = 0 \\ - x, x < 0\end{cases}\]
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
For what value of k is the function
\[f\left( x \right) = \begin{cases}\frac{\sin 5x}{3x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0?\]
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.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\left| x - 3 \right|, & \text{ if } x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & \text{ if } x < 1\end{cases}\]
Find f (0), so that \[f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}\] becomes continuous at x = 0.
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 \[f\left( x \right) = \begin{cases}a \sin\frac{\pi}{2}\left( x + 1 \right), & x \leq 0 \\ \frac{\tan x - \sin x}{x^3}, & x > 0\end{cases}\] is continuous at x = 0, then a equals
Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.
Discuss the continuity and differentiability of f (x) = |log |x||.
Give an example of a function which is continuos but not differentiable at at a point.
Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]
If \[f\left( x \right) = \left| \log_e |x| \right|\]
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\] then f (x) is
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
Find `dy/dx if y = tan^-1 ((6x)/[ 1 - 5x^2])`
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
Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if" x ≠ 0),(0",", "if" x = 0):}` is discontinuous at x = 0.
The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.
The function f(x) = |x| + |x – 1| is ______.
f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2
f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1
f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",", "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",", "if" 0 ≤ x ≤ 1):}` 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
The composition of two continuous function is a continuous function.
The value of k (k < 0) for which the function f defined as
f(x) = `{((1-cos"kx")/("x"sin"x")"," "x" ≠ 0),(1/2"," "x" = 0):}`
is continuous at x = 0 is:
Given functions `"f"("x") = ("x"^2 - 4)/("x" - 2) "and g"("x") = "x" + 2, "x" le "R"`. Then which of the following is correct?
