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.
For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if" x <= 0),(4x+ 1", if" x > 0):}` continuous at x = 0? What about continuity at x = 1?
Discuss the continuity of the following function:
f(x) = sin x × cos x
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
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
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.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}(x - 1)\tan\frac{\pi x}{2}, \text{ if } & x \neq 1 \\ k , if & x = 1\end{cases}\] at x = 1at x = 1
If \[f\left( x \right) = \begin{cases}\frac{x^2}{2}, & \text{ if } 0 \leq x \leq 1 \\ 2 x^2 - 3x + \frac{3}{2}, & \text P{ \text{ if } } 1 < x \leq 2\end{cases}\]. Show that f is continuous at x = 1.
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.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & \text{ if } x < 0 \\ 2x + 3, & x \geq 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, & \text{ if } x \neq 0 \\ 3k , & \text{ if } x = 0\end{cases}\]
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.
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 \[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, write the value of k.
\[f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}\] then f (x) is continuous for all
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
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 \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
The function f (x) = 1 + |cos x| 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 ______.
`lim_("x" -> 0) (1 - "cos" 4 "x")/"x"^2` is equal to ____________.
Let f(x) = `{{:(5^(1/x), x < 0),(lambda[x], x ≥ 0):}` and λ ∈ R, then at x = 0
The function f(x) = 5x – 3 is continuous at x =
What is the values of' 'k' so that the function 'f' is continuous at the indicated point
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx^2",", if x ≤ 2),(3",", if x > 2):}` at x = 2
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx + 1",", if x ≤ pi),(cos x",", if x > pi):}` at = `pi`
The function f(x) = x |x| is ______.
Discuss the continuity of the following function:
f(x) = sin x + cos x
For \[\frac{f}{g}\] to be continuous at \[x=c\], which condition is required?
What is the definition of the composite function \[f\circ g\]?
Why is \[g(x)=1-x+|x|\] continuous?
