Question

Determine if the given function is continuous at a given value of x. Show the solution.

Answer 1

This question is taken from the continuity and discontinuity in which we have to check whether a given function is continuous at the given point or not continuous, so we can write the given function below

`f(x)=(3x)/(x^2-4)`

at x=4 is continuous or not.

First, we are written the given function and check whether it is continuous at a given point or not, so we have to check the left hand and right-hand limit (LHL and RHL)are equal or not, Given a function

`f(x)=(3x)/(x^2-4)`

so formula we write it below at the given point

LHL=limx→−4−k(x)

RHL=limx→−4+k(x)

and applicable only limits are exist otherwise discontinuous.

RHL=LHL=k(4)

if the above formula is satisfy then the given function is continuous

now we calculate the L.H.L

LHL=limx→−4−k(x)

`\text{LHL}=\lim _(x\rightarrow-4^-)(3x)/(x^2-4)`

Now change the limit, x=-4-h and h=0.

`LHL=\lim _(h\rightarrow0)(3(-4-h))/((-4-h)^2-4)=(-12)/(12)=-1`

For RHL,

`\text{RHL}=\lim _(h\rightarrow0)(3(4-h))/((4-h)^2-4)=(12)/(12)=1`

For x=4,

`f(4)=(3*4)/(4^2-4)=(12)/(12)=1`

So RHL is not equal to LHL but RHL = for the function at x=4.

Then it is not satisfied the continuous property so the function is discontinuous at x=4.

So its function value is discontinuous at x= 4.