Advertisements
Advertisements
Question
The function \[f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}\]
Options
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
Solution
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
RELATED QUESTIONS
Examine the following function for continuity:
f(x) = | x – 5|
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.
Discuss the continuity of the following functions at the indicated point(s):
Discuss the continuity of the function f(x) at the point x = 0, where \[f\left( x \right) = \begin{cases}x, x > 0 \\ 1, x = 0 \\ - x, x < 0\end{cases}\]
If \[f\left( x \right) = \begin{cases}\frac{x - 4}{\left| x - 4 \right|} + a, \text{ if } & x < 4 \\ a + b , \text{ if } & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, \text{ if } & x > 4\end{cases}\] is continuous at x = 4, find a, b.
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
Find all point of discontinuity of the function
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 16}{x - 4}, & \text{ if } x \neq 4 \\ k , & \text{ if } x = 4\end{cases}\] is continuous at x = 4, find k.
Write the value of b for which \[f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}\] is continuous at x = 1.
The function
If \[f\left( x \right) = \begin{cases}\frac{{36}^x - 9^x - 4^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, then k equals
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 value of k which makes \[f\left( x \right) = \begin{cases}\sin\frac{1}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] continuous at x = 0, is
The points of discontinuity of the function\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]
Show that the function
(i) differentiable at x = 0, if m > 1
(ii) continuous but not differentiable at x = 0, if 0 < m < 1
(iii) neither continuous nor differentiable, if m ≤ 0
Write the points where f (x) = |loge x| is not differentiable.
The function f (x) = sin−1 (cos x) is
The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is
Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is
Find the value of k for which the function f (x ) = \[\binom{\frac{x^2 + 3x - 10}{x - 2}, x \neq 2}{ k , x^2 }\] is continuous at x = 2 .
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.
Find `dy/dx if y = tan^-1 ((6x)/[ 1 - 5x^2])`
Examine the continuity of the followin function :
`{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`
If the function
f(x) = x2 + ax + b, x < 2
= 3x + 2, 2≤ x ≤ 4
= 2ax + 5b, 4 < x
is continuous at x = 2 and x = 4, then find the values of a and b
Examine the differentiability of the function f defined by
f(x) = `{{:(2x + 3",", "if" -3 ≤ x < - 2),(x + 1",", "if" -2 ≤ x < 0),(x + 2",", "if" 0 ≤ x ≤ 1):}`
The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.
y = |x – 1| is a continuous function.
Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1
f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2
Find the values of a and b such that the function f defined by
f(x) = `{{:((x - 4)/(|x - 4|) + "a"",", "if" x < 4),("a" + "b"",", "if" x = 4),((x - 4)/(|x - 4|) + "b"",", "if" x > 4):}`
is a continuous function at x = 4.
Examine the differentiability of f, where f is defined by
f(x) = `{{:(1 + x",", "if" x ≤ 2),(5 - x",", "if" x > 2):}` at x = 2
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
Write the number of points where f(x) = |x + 2| + |x - 3| is not differentiable.
