Question

Examine the continuity of the following:

`sinx/x^2`

Answer 1

f(x) = `sinx/x^2`

f(x) is not defined at x = 0

∴ f(x) is defined for all points of R – {0}

Let x₀ be an arbitrary point in R – {0}.

Then `lim_(x -> x_0) f(x) =  lim_(x -> x_0) sinx/x^2`

= `(sin x_0)/(x_0^2)`  .......(1)

`f(x_0) = (sin x_0)/(x_0^2)`  .......(2)

From equation (1) and (2) we have

`lim_(x -> x_0) sinx/x^2 = f(x_0)`

∴ The limit of the function f(x) exist at x = x₀ and is equal to the value of the function f(x) at x = x₀.

Since x₀ is an arbitrary point in R – {0}, the above result is true for all points in R – {0}.

∴ f(x) is continuous at all points of R – {0}.