Advertisements
Advertisements
Question
Examine the following function for continuity:
f(x) = |x – 5|
Advertisements
Solution
Let f(x) = |x – 5|
`lim_(x->a^+)` f(x) = `lim_(h->0)` |a + h − 5|
= |a − 5|
= a − 5
`lim_(x->a^-)` f(x) = `lim_(h->0)` |a − h − 5|
= |a − 5|
= a − 5
f(a) = |a − 5| = a − 5
∴ `lim_(x->a^+)` f(x) = `lim_(x->a^-)` f(x) = f(a)
Hence, the given function f(x) = |x − 5| is continuous.
APPEARS IN
RELATED QUESTIONS
Examine the following function for continuity:
f(x) = `1/(x - 5)`, 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):}`
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(-2", if" x <= -1),(2x", if" -1 < x <= 1),(2", if" x > 1):}`
Show that
Find the value of 'a' for which the function f defined by
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 \[f\left( x \right) = \begin{cases}\frac{1 - \cos kx}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\text{is continuous at x} = 0, \text{ find } k .\]
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:
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}\frac{\sqrt{1 + px} - \sqrt{1 - px}}{x}, & \text{ if } - 1 \leq x < 0 \\ \frac{2x + 1}{x - 2} , & \text{ if } 0 \leq x \leq 1\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}5 , & \text{ if } & x \leq 2 \\ ax + b, & \text{ if } & 2 < x < 10 \\ 21 , & \text{ if } & x \geq 10\end{cases}\]
The function
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 \[f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}\]is continuous at x = 0, then
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\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]
If f is defined by f (x) = x2, find f'(2).
Discuss the continuity and differentiability of
Give an example of a function which is continuos but not differentiable at at a point.
Write the points where f (x) = |loge x| is not differentiable.
If \[f\left( x \right) = \sqrt{1 - \sqrt{1 - x^2}},\text{ then } f \left( x \right)\text { is }\]
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
Evaluate :`int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx`
`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 f(x) = `(e^(2x) - 1)/(ax)` . for x < 0 , a ≠ 0
= 1. for x = 0
= `(log(1 + 7x))/(bx)`. for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b
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
Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1
= `-1/3` for x = 1, at x = 1
Discuss the continuity of the function f(x) = sin x . cos x.
The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.
The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.
The value of k which makes the function defined by f(x) = `{{:(sin 1/x",", "if" x ≠ 0),("k"",", "if" x = 0):}`, continuous at x = 0 is ______.
f(x) = `{{:((1 - cos 2x)/x^2",", "if" x ≠ 0),(5",", "if" x = 0):}` at x = 0
f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2
f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.
The composition of two continuous function is a continuous function.
