Advertisements
Advertisements
प्रश्न
Discuss the continuity of the following functions at the indicated point(s):
Advertisements
उत्तर
Given:
Here,
\[\lim_{x \to 1} f\left( x \right) = \lim_{x \to 1} \frac{1 - x^n}{1 - x}\]
\[ \Rightarrow \lim_{x \to 1} f\left( x \right) = \lim_{x \to 1} \left[ \left( 1 - x \right)^{n - 1} + \ prescript{n}{}{C}_1 \left( 1 - x \right)^{n - 2} x + \ prescript{n}{}{C}_2 \left( 1 - x \right)^{n - 3} x^2 + . . . + \ prescript{n}{}{C}_{n - 1} \left( 1 - x \right)^0 x^{n - 1} \right]\]
Thus,
APPEARS IN
संबंधित प्रश्न
If 'f' is continuous at x = 0, then find f(0).
`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`
Examine the following function for continuity:
f(x) = x – 5
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(3", if" 0 <= x <= 1),(4", if" 1 < x < 3),(5", if" 3 <= x <= 10):}`
A function f(x) is defined as,
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}; for & x \neq 1 \\ 2 ; for & x = 1\end{cases}\] Find whether f(x) is continuous at x = 1.
Show that
is discontinuous at x = 0.
Discuss the continuity of the following functions at the indicated point(s):
Discuss the continuity of the following functions at the indicated point(s):
Find the value of 'a' for which the function f defined by
Discuss the continuity of the function f(x) at the point x = 0, where \[f\left( x \right) = \begin{cases}x, x > 0 \\ 1, x = 0 \\ - x, x < 0\end{cases}\]
Discuss the continuity of \[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0\]
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
Find the value of k for which \[f\left( x \right) = \begin{cases}\frac{1 - \cos 4x}{8 x^2}, \text{ when} & x \neq 0 \\ k ,\text{ when } & x = 0\end{cases}\] is continuous at x = 0;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}\]at x = 5
If the functions f(x), defined below is continuous at x = 0, find the value of k. \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2x}{2 x^2}, & x < 0 \\ k , & x = 0 \\ \frac{x}{\left| x \right|} , & x > 0\end{cases}\]
Find the points of discontinuity, if any, of the following functions:
Find the values of a and b so that the function f(x) defined by \[f\left( x \right) = \begin{cases}x + a\sqrt{2}\sin x , & \text{ if }0 \leq x < \pi/4 \\ 2x \cot x + b , & \text{ if } \pi/4 \leq x < \pi/2 \\ a \cos 2x - b \sin x, & \text{ if } \pi/2 \leq x \leq \pi\end{cases}\]becomes continuous on [0, π].
Discuss the continuity of the function \[f\left( x \right) = \begin{cases}2x - 1 , & \text { if } x < 2 \\ \frac{3x}{2} , & \text{ if } x \geq 2\end{cases}\]
Let \[f\left( x \right) = \begin{cases}\frac{x^4 - 5 x^2 + 4}{\left| \left( x - 1 \right) \left( x - 2 \right) \right|}, & x \neq 1, 2 \\ 6 , & x = 1 \\ 12 , & x = 2\end{cases}\]. Then, f (x) is continuous on the set
If \[f\left( x \right) = \begin{cases}mx + 1 , & x \leq \frac{\pi}{2} \\ \sin x + n, & x > \frac{\pi}{2}\end{cases}\] is continuous at \[x = \frac{\pi}{2}\] , then
If \[f\left( x \right) = \frac{1}{1 - x}\] , then the set of points discontinuity of the function f (f(f(x))) is
If \[f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.\] then the value of (a, b) for which f (x) cannot be continuous at x = 1, is
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 \[f\left( x \right) = \begin{cases}\frac{\sin \left( \cos x \right) - \cos x}{\left( \pi - 2x \right)^2}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}\]is continuous at x = π/2, then k is equal to
Discuss the continuity and differentiability of f (x) = e|x| .
If \[f\left( x \right) = \left| \log_e |x| \right|\]
Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4
= 10 for x = 4 at x = 4
Discuss the continuity of f at x = 1 ,
Where f(x) = `(3 - sqrt(2x + 7))/(x - 1)` for x = ≠ 1
= `(-1)/3` for x = 1
If f is continuous at x = 0, then find f (0).
Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`
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.
Examine the continuity of the followin function :
`{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`
Discuss the continuity of the function f(x) = sin x . cos x.
Show that the function f defined by f(x) = `{{:(x sin 1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.
The function given by f (x) = tanx is discontinuous on the set ______.
Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))
`lim_("x" -> "x" //4) ("cos x - sin x")/("x"- "x" /4)` is equal to ____________.
Write the number of points where f(x) = |x + 2| + |x - 3| is not differentiable.
