Show that is continuous at x = 0

__Ideas required to solve the problem:__

1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x) , we can say that if we plot the coordinates (x , f(x)) and try to join all those points in the specified region, we can do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve without any breakage.

Mathematically we define the same thing as given below:

A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to be checked

If:–

equation 1

where h is a very small positive no (can assume h = 0.00000000001 like this )

It means :–

Limiting value of the left neighbourhood of x = c also called left hand limit LHL must be equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both must be equal to the value of f(x) at x=c i.e. f(c).

Thus, it is the necessary condition for a function to be continuous

So, whenever we check continuity we try to check above equality if it holds true, function is continuous else it is discontinuous.

2. Idea of sandwich theorem – This theorem also known as squeeze theorem that you may have encountered in your class 11 in limits chapter suggests that

If I be an interval having the point a as a limit point. Let g, f, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have{displaystyle lim _{x→ a}g(x)=lim _{x→ a}h(x)=L.}{displaystyle lim _{x→ a}f(x)=L.}

g(x) ≤ f(x) ≤ h(x) and also

Then , We say that f(x) is squeezed between g(x) and h(x) or you can assume it like sandwich.

∴ …… equation 2

…… equation 3

NOTE : denominator in the above limits should be exactly same as that of content in sine function

Eg :

3. Idea of logarithmic limit and exponential limit –

……equation 4

…… equation 5

You must have read such limits in class 11. You can verify these by expanding log(1+x) and e^{x} in its taylor form.

Numerator and denominator conditions also hold for this limit like sandwich theorem.

E.g :

But,

Now we are ready to solve this problem:

Given function is :

…… equation 6

As we need to check continuity at x=0 , so we need to check value of f(x) at x = 0,left hand and right hand limits. If all 3 comes out to be equal we can say that f(x) is continuous at x=0 else it is discontinuous.

Clearly,

LHL =

Putting x = –h in equation 6, we have

LHL = [∵ sin(–θ) = sinθ and also tan(–θ) = tanθ ]

= [using equations 2 and 3 to apply sandwich theorem]

RHL =

Putting x = h in equation 6, we have

RHL = = [trying to apply logarithmic and exponential limit using equation 4 and 5]

= =

Clearly, LHL = RHL = f(0) =

∴ f(x) is continuous at x = 0.

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