Advertisements
Advertisements
प्रश्न
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
उत्तर
Since f is continuous at x = 0.
`lim_( x -> 0 ) f(x) = f(0) `
Given f(0) = k
∴ `lim_( x -> 0) f(x) = k` (i)
Now `lim_( x -> 0) f(x) = lim_( x -> 0 ) [(e^x - 1)sinx]/[x^2]`
= `lim_( x -> 0) ([e^x - 1]/[x])([sin x]/[x])`
= `lim_( x -> 0) ([e^x - 1]/[x]) lim_(x->0)([sin x]/[x])`
= log e x 1
= 1 x 1
`therefore lim_( x -> 0 ) = f(x) = 1` (ii)
from (i) and (ii)
k = 1
संबंधित प्रश्न
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.
Discuss the continuity of the following functions at the indicated point(s):
(ii) \[f\left( x \right) = \left\{ \begin{array}{l}x^2 \sin\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]
Find the value of 'a' for which the function f defined by
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; \[f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}\]at x = 1
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\left| x - 3 \right|, & \text{ if } x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & \text{ if } x < 1\end{cases}\]
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.
If \[f\left( x \right) = \left| \log_{10} x \right|\] then at x = 1
The values of the constants a, b and c for which the function \[f\left( x \right) = \begin{cases}\left( 1 + ax \right)^{1/x} , & x < 0 \\ b , & x = 0 \\ \frac{\left( x + c \right)^{1/3} - 1}{\left( x + 1 \right)^{1/2} - 1}, & x > 0\end{cases}\] may be continuous at x = 0, are
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
Write the points where f (x) = |loge x| is not differentiable.
Let f (x) = |x| and g (x) = |x3|, then
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
If y = ( sin x )x , Find `dy/dx`
Discuss the continuity of function f at x = 0.
Where f(X) = `[ [sqrt ( 4 + x ) - 2 ]/ ( 3x )]`, For x ≠ 0
= `1/12`, For x = 0
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
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
Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.
