Advertisements
Advertisements
प्रश्न
Discuss the continuity of f(x) = sin | x |.
Advertisements
उत्तर
`" Let " f(x)=sin |x| `
This function f is defined for every real number and f can be written as the composition of two functions as, f = h o g, where g (x) = |x| and h (x) = sin x
\[\left[ \because hog\left( x \right) = h\left( g\left( x \right) \right) = h\left( \left| x \right| \right) = \sin \left| x \right| \right]\]
It has to be proved first that `g(x)=|x|` and `h(x)=sinx` are continous function .
`g(x)=|x|`can be written as
`g(x)=[[-x, if x < 0],[x, if x≥ 0]]`
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
`if c < 0 " then " g(c)=-c ` and `lim_(x->c)g(x)=lim_(x->c)(-x)=-c`
`∴lim_(x->c) g (x) = g(c)`
Therefore, g is continuous at all points x > 0
Case III:
` " If " c=0, `then `g(c)=g(0)=0`
`lim_(x->0)g(x)=lim_(x->0)(x)=0`
`lim_(x->0)g(x)=lim_(x->0)(x)=0`
`∴lim_(x->0)g(x)=lim(x)=g(0)`
Therefore, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all points.
Now, h (x) = sin x
It is evident that h (x) = sin x is defined for every real number.
Let c be a real number.
Put x = c + k
If x → c, then k → 0
h (c) = sin c
`h(c)=sin c`
`lim_(x->c)(x)=lim_(x->c)sin x`
`=lim_(k->0)sin(c+k)`
`=lim_(k->0)[sin c cos k + cos c sin k]`
`= lim_(k->0)(sin c cos k)+lim_(k->0)(cos c sin k)`
`=sin c cos 0 + cos c sin 0`
`=sin c+0`
`=sin c`
`∵lim_(x->c)h(x)=g(c)`
So, h is a continuous function.
APPEARS IN
संबंधित प्रश्न
Find the relationship between a and b so that the function f defined by f(x) = `{(ax + 1", if" x<= 3),(bx + 3", if" x > 3):}` is continuous at x = 3.
Is the function defined by f(x) = x2 − sin x + 5 continuous at x = π?
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{((kcosx)/(pi-2x)", if" x != pi/2),(3", if" x = pi/2):}` at x = `"pi/2`
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx^2", if" x<= 2),(3", if" x > 2):}` at x = 2
Examine that sin |x| is a continuous function.
Examine the continuity of the function
\[f\left( x \right) = \left\{ \begin{array}{l}3x - 2, & x \leq 0 \\ x + 1 , & x > 0\end{array}at x = 0 \right.\]
Also sketch the graph of this function.
Find the values of a so that the function
Let \[f\left( x \right) = \frac{\log\left( 1 + \frac{x}{a} \right) - \log\left( 1 - \frac{x}{b} \right)}{x}\] x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
If \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin } x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (0).
Prove that the function \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0\end{cases}\] is everywhere continuous.
Discuss the continuity of the function
Find the points of discontinuity, if any, of the following functions:
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
Show that f (x) = cos x2 is a continuous function.
Show that f (x) = | cos x | is a continuous function.
If \[f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then write the value of k.
Determine whether \[f\left( x \right) = \binom{\frac{\sin x^2}{x}, x \neq 0}{0, x = 0}\] is continuous at x = 0 or not.
If \[f\left( x \right) = \begin{cases}\frac{\log\left( 1 + ax \right) - \log\left( 1 - bx \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] and f (x) is continuous at x = 0, then the value of k is
The function \[f\left( x \right) = \begin{cases}1 , & \left| x \right| \geq 1 & \\ \frac{1}{n^2} , & \frac{1}{n} < \left| x \right| & < \frac{1}{n - 1}, n = 2, 3, . . . \\ 0 , & x = 0 &\end{cases}\]
If \[f\left( x \right) = \frac{1 - \sin x}{\left( \pi - 2x \right)^2},\] when x ≠ π/2 and f (π/2) = λ, then f (x) will be continuous function at x= π/2, where λ =
The value of a for which the function \[f\left( x \right) = \begin{cases}\frac{\left( 4^x - 1 \right)^3}{\sin\left( x/a \right) \log \left\{ \left( 1 + x^2 /3 \right) \right\}}, & x \neq 0 \\ 12 \left( \log 4 \right)^3 , & x = 0\end{cases}\]may be continuous at x = 0 is
The function
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos 10x}{x^2} , & x < 0 \\ a , & x = 0 \\ \frac{\sqrt{x}}{\sqrt{625 + \sqrt{x}} - 25}, & x > 0\end{cases}\] then the value of a so that f (x) may be continuous at x = 0, is
Find the values of a and b so that the function
If f is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\]
then at x = 0, f (x) is
If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.
The function f(x) = `"e"^|x|` is ______.
Let f(x) = |sin x|. Then ______.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
What is the values of' 'k' so that the function 'f' is continuous at the indicated point
The function f(x) = x |x| is ______.
