Advertisements
Advertisements
प्रश्न
The function \[f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}\]
विकल्प
is continuous at x = 0
is not continuous at x = 0
is not continuous at x = 0, but can be made continuous at x = 0
none of these
उत्तर
is not continuous at x = 0
Given:
We have
If \[e^\frac{1}{x} = t\] , then
\[x \to 0, t \to \infty\]
Also,
\[\therefore \lim_{x \to 0} f\left( x \right) \neq f\left( 0 \right)\]
Hence,
APPEARS IN
संबंधित प्रश्न
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) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`
If \[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\]
Find whether f(x) is continuous at x = 0.
If \[f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}\] find whether f is continuous at x = 0.
Let \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x^2}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\] Show that f(x) is discontinuous at x = 0.
Discuss the continuity of the following functions at the indicated point(s): (iv) \[f\left( x \right) = \left\{ \begin{array}{l}\frac{e^x - 1}{\log(1 + 2x)}, if & x \neq a \\ 7 , if & x = 0\end{array}at x = 0 \right.\]
For what value of k is the function
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}\] at x = 5
Let\[f\left( x \right) = \left\{ \begin{array}\frac{1 - \sin^3 x}{3 \cos^2 x} , & \text{ if } x < \frac{\pi}{2} \\ a , & \text{ if } x = \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x )^2}, & \text{ if } x > \frac{\pi}{2}\end{array} . \right.\] ]If f(x) is continuous at x = \[\frac{\pi}{2}\] , find a and b.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\left| x - 3 \right|, & \text{ if } x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & \text{ if } x < 1\end{cases}\]
Find f (0), so that \[f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}\] becomes continuous at x = 0.
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
The value of f (0), so that the function
The points of discontinuity of the function\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]
Find whether the function is differentiable at x = 1 and x = 2
Define differentiability of a function at a point.
Let f (x) = |x| and g (x) = |x3|, then
If \[f\left( x \right) = \left| \log_e |x| \right|\]
If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\] then f (x) is
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`
Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`
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 following function :
`{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4`
Examine the continuity of the following function :
f(x) = x2 - x + 9, for x ≤ 3
= 4x + 3, for x > 3
at x = 3.
If y = ( sin x )x , Find `dy/dx`
Discuss the continuity of the function f at x = 0, where
f(x) = `(5^x + 5^-x - 2)/(cos2x - cos6x),` for x ≠ 0
= `1/8(log 5)^2,` for x = 0
The function given by f (x) = tanx is discontinuous on the set ______.
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 ______.
y = |x – 1| is a continuous function.
f(x) = `{{:(|x - 4|/(2(x - 4))",", "if" x ≠ 4),(0",", "if" x = 4):}` at x = 4
f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",", "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",", "if" 0 ≤ x ≤ 1):}` at x = 0
Prove that the function f defined by
f(x) = `{{:(x/(|x| + 2x^2)",", x ≠ 0),("k", x = 0):}`
remains discontinuous at x = 0, regardless the choice of k.
Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",", "if" x ≤ 1),("q"x + 2",", "if" x > 1):}` is differentiable at x = 1
`lim_("x" -> "x" //4) ("cos x - sin x")/("x"- "x" /4)` is equal to ____________.
