Advertisements
Advertisements
प्रश्न
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
Advertisements
उत्तर
Given:
We have
(LHL at x = 0) =
(RHL at x = 0) =
Also
Now,
(LHL at x = 1) =
(RHL at x =1) =
Also,
Hence,
APPEARS IN
संबंधित प्रश्न
Examine the following function for continuity:
f(x) = `1/(x - 5)`, x ≠ 5
Examine the following function for continuity:
f(x) = |x – 5|
If \[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\]
Find whether f(x) is continuous at x = 0.
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.
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.
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos kx}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\text{is continuous at x} = 0, \text{ find } k .\]
For what value of k is the function
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; \[f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}\] at x = 5
Let\[f\left( x \right) = \left\{ \begin{array}\frac{1 - \sin^3 x}{3 \cos^2 x} , & \text{ if } x < \frac{\pi}{2} \\ a , & \text{ if } x = \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x )^2}, & \text{ if } x > \frac{\pi}{2}\end{array} . \right.\] ]If f(x) is continuous at x = \[\frac{\pi}{2}\] , find a and b.
Find the points of discontinuity, if any, of the following functions:
Find all point of discontinuity of the function
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 16}{x - 4}, & \text{ if } x \neq 4 \\ k , & \text{ if } x = 4\end{cases}\] is continuous at x = 4, find k.
If \[f\left( x \right) = \left| \log_{10} x \right|\] then at x = 1
The function \[f\left( x \right) = \frac{x^3 + x^2 - 16x + 20}{x - 2}\] is not defined for x = 2. In order to make f (x) continuous at x = 2, Here f (2) should be defined as ______.
The points of discontinuity of the function
\[f\left( x \right) = \begin{cases}2\sqrt{x} , & 0 \leq x \leq 1 \\ 4 - 2x , & 1 < x < \frac{5}{2} \\ 2x - 7 , & \frac{5}{2} \leq x \leq 4\end{cases}\text{ is } \left( \text{ are }\right)\]
If \[f\left( x \right) = \begin{cases}\frac{1 - \sin^2 x}{3 \cos^2 x} , & x < \frac{\pi}{2} \\ a , & x = \frac{\pi}{2} \\ \frac{b\left( 1 - \sin x \right)}{\left( \pi - 2x \right)^2}, & x > \frac{\pi}{2}\end{cases}\]. Then, f (x) is continuous at \[x = \frac{\pi}{2}\], if
Define differentiability of a function at a point.
Give an example of a function which is continuos but not differentiable at at a point.
Write the points of non-differentiability of
Let f (x) = |sin x|. Then,
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 whether the following function is differentiable at x = 1 and x = 2 or not : \[f\left( x \right) = \begin{cases}x, & & x < 1 \\ 2 - x, & & 1 \leq x \leq 2 \\ - 2 + 3x - x^2 , & & x > 2\end{cases}\] .
`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 the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.
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)`
If the function
f(x) = x2 + ax + b, x < 2
= 3x + 2, 2≤ x ≤ 4
= 2ax + 5b, 4 < x
is continuous at x = 2 and x = 4, then find the values of a and b
If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)` so that f (x) becomes continuous at x = `pi/4`
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.
f(x) = `{{:((1 - cos 2x)/x^2",", "if" x ≠ 0),(5",", "if" x = 0):}` at x = 0
f(x) = `{{:(|x - 4|/(2(x - 4))",", "if" x ≠ 4),(0",", "if" x = 4):}` at x = 4
f(x) = `{{:(|x - "a"| sin 1/(x - "a")",", "if" x ≠ 0),(0",", "if" x = "a"):}` at x = a
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
If f is continuous on its domain D, then |f| is also continuous on D.
