Advertisements
Advertisements
प्रश्न
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Advertisements
उत्तर
(i) f(x) = cos (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (cos c) = (cos c) = (cos c)
This statement is true; that is, f(x) is continuous at every point on the real number line.
(ii) f(x) = cosec (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x ->^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (cosec c) = (cosec c) = (cosec c)
This statement is true; that is, f(x) is continuous at every point on the real number line.
(iii) f(x) = sec (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (sec c) = (sec c) = (sec c)
This statement is true; that is, f(x) is continuous at every point on the real number line.
(iv) f(x) = cot (x)
Let c be any real number such that (n − 1)π < x < nπ, where n represents an integer point.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ (cot c) = (cot c) = (cot c)
This statement is true; that is, f(x) is continuous at every point on the real number line between (n − 1)π and nπ.
Now if we consider c such that c = nπ, where n represents an integer point, then:
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ ±∞ = ±∞ = ±∞
That is, f(x) is continuous at every point on the real number line except at the nπ-type points.
APPEARS IN
संबंधित प्रश्न
If f (x) is continuous on [–4, 2] defined as
f (x) = 6b – 3ax, for -4 ≤ x < –2
= 4x + 1, for –2 ≤ x ≤ 2
Show that a + b =`-7/6`
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?
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) = `{(kx^2", if" x<= 2),(3", if" x > 2):}` at x = 2
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx + 1", if" x <= 5),(3x - 5", if" x > 5):}` at x = 5
Show that the function defined by f(x) = |cos x| is a continuous function.
Determine the value of the constant k so that the function
\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0 .\]
Find the values of a so that the function
Find the value of k if f(x) is continuous at x = π/2, where \[f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x}, & x \neq \pi/2 \\ 3 , & x = \pi/2\end{cases}\]
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.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & \text{ if } x \neq 2 \\ 16 , & \text{ if } x = 2\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & \text{ if } x < 0 \\ 2x + 3, & x \geq 0\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
If \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}\]
for x ≠ π/4, find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].
Show that f (x) = cos x2 is a continuous function.
What happens to a function f (x) at x = a, if
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 \[f\left( x \right) = \frac{1 - \sin x}{\left( \pi - 2x \right)^2},\] 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
Let \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .\] The value which should be assigned to f (x) at \[x = \frac{\pi}{4},\]so that it is continuous everywhere is
The value of a for which the function \[f\left( x \right) = \begin{cases}5x - 4 , & \text{ if } 0 < x \leq 1 \\ 4 x^2 + 3ax, & \text{ if } 1 < x < 2\end{cases}\] is continuous at every point of its domain, is
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
Let f (x) = |cos x|. Then,
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) = `(4 - x^2)/(4x - x^3)` is ______.
The value of f(0) for the function `f(x) = 1/x[log(1 + x) - log(1 - x)]` to be continuous at x = 0 should be
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
Which statement about \[\sin x\] and \[\cos x\] is correct?
If \[f\] is continuous, which scalar multiple is continuous?
For \[f(x)=|1-x+|x||\], which functions give the representation \[f(x)=h(g(x))\]?
Why is \[f(x)=|1-x+|x||\] continuous for every real \[x\]?
