Question

Examine the continuity of the following:

cot x + tan x

Answer 1

Let f(x) = cot x + tan x

f(x) is not defined a x= `("n"x)/2`, n ∈ z

∴ f(x) is defined for all pints of `"R" - {("n"pi)/2, "n" ∈ "z"}` 

Let x₀ be an arbitrary point in `"R" - {("n"pi)/2}`, n ∈ z

Then `lim_(x -> x_0) f(x) =  lim_(x -> x_0) (cotx + tan x)`

= `cox_ + tan x_0`  .......(1)

`f(x_0) = cot x_0  +  tan x_0` .......(2) 

From equation (1) and (2) we have

`lim_(x -> x_0) (cotx + tan x) = f(x_0)`

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

Since x₀ is an arbitrary point , the above result is true for all points of `"R" - {("n"x)/2}`, n ∈ z.

∴ f(x) is continuous at all points of `"R" - {("n"x)/2}`, n ∈ z.