Question

Examine the continuity of the following:

e^x tan x

Answer 1

Let f(x) = e^x tan x

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

Expect at `(2"n" + 1) pi/2`, n ∈ Z.

Let x₀ be an arbitrary point in `"R" - (2"n" + 1) pi/2`, n ∈ Z

Then `lim_(x -> x_0) f(x) =  lim_(x -> x_0) "e"^x tan x`

= `"e"^(x_0)  tan x_0`  .......(1)

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

From equation (1) and (2) we get

`lim_(x -> x_0)  "e"^x  tan x = f(x_0)`

∴ 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. f(x) exists at all points in `"R" - (2"n" + 1) pi/2`, n ∈ Z and is equal to the value of the function f(x) at that points.

∴ f(x) satisfies all conditions for continuity.

Hence, f(x) is continuous at all points of `"R" - (2"n" + 1) pi/2`, n ∈ Z