Question

Examine the continuity of the following:

e^2x + x²

Answer 1

Let f(x) = e^2x + x²

Clearly, f(x) is defined for all points in R.

Let x₀ be an arbitrary point in R.

`lim_(x -> x_0) f(x) =  lim_(x -> x0) ("e"(2x) + x^2)`

= `"e"^(2x_0) + x_0^2`  ........(1)

`f(x_0) = "e"^(2x_0) + x_0^2`  ........(2)

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

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, the above is true for all points in R.

Hence f(x) satisfies all conditions for continuity.

Hence f(x) is continuous at all points of R.