Question

Examine the continuity of the following:

x² cos x

Answer 1

Let f(x) = x² cos x

f(x) is defined at all points of R.

Let x₀ be an arbitrary point in R.

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

= x²₀ cos ₀

f(x₀) = x²₀ cos ₀

From equation (1) and (2), we have

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

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

Since x₀ is arbitrary, the limit of the function exist and is equal to the value of the function for all points in R.

∴ f(x) satisfies all conditions for continuity.

Hence f (x) is a continuous function in R.