Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Find the value of 'k' if the function
`f(X)=(tan7x)/(2x) , "for " x != 0 `
`=k`, for x=0
is continuos at x=0
Discuss the continuity of the following functions at the indicated point(s):
(ii) \[f\left( x \right) = \left\{ \begin{array}{l}x^2 \sin\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]
Discuss the continuity of \[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0\]
For what value of k is the following function continuous at x = 1? \[f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k , & x = 1\end{cases}\]
If the functions f(x), defined below is continuous at x = 0, find the value of k. \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2x}{2 x^2}, & x < 0 \\ k , & x = 0 \\ \frac{x}{\left| x \right|} , & x > 0\end{cases}\]
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
Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
If the function f is continuous at x = 0
Where f(x) = 2`sqrt(x^3 + 1)` + a, for x < 0,
= `x^3 + a + b, for x > 0
and f (1) = 2, then find a and b.
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 at the point given. If the function is discontinuous, then remove the discontinuity.
f (x) = `(sin^2 5x)/x^2` for x ≠ 0
= 5 for x = 0, at x = 0
If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)` so that f (x) becomes continuous at x = `pi/4`
The function given by f (x) = tanx is discontinuous on the set ______.
The function f(x) = |x| + |x – 1| is ______.
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.
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
`lim_("x" -> 0) (2 "sin x - sin" 2 "x")/"x"^3` is equal to ____________.
