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प्रश्न
Examine the continuity of the following:
cot x + tan x
उत्तर
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.
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