Question

if f(x)= xcosx+since/x when x is not equal to 0​

Answer 1

Answer:

k=1

this will be the solution

Answer 2

$\huge\mathcal{Answer࿐}$

Explain : -

To prove that a function is continuous at a given point, let's say the point is a.

Now,

LHL = RHL = f(a)

If the condition is true then it is said that the function is continuous at point a.

Given function,

f(x) = xcos(1/x)

To check continuity at x = 0

First we will calculate LHL(Left Hand Limit)

limlim x−>0−xcos(1/x)x−>0−xcos(1/x)

As x−>0x−>0 …. 1/x−>1/x−> −infinity−infinity

And cos(1/x)cos(1/x) will oscillate between (−1,1)(−1,1)

SinceSince limlim x−>0−cos(1/x)x−>0−cos(1/x) is a finite quantity thus

limlim x−>0−xcos(1/x)=0x−>0−xcos(1/x)=0

Therefore LHL = 0

For RHL(Right Hand Limit)

By applying same approach

limlim x−>0+xcos(1/x)=0x−>0+xcos(1/x)=0

This gives RHL = 0

AND it is given that f(0) = 0

This gives LHL = RHL = f(0) = 0

Therefore the function,

f(x) = xcos(1/x) is continuous at x=0