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Question
Examine the continuity of the following:
ex tan x
Solution
Let f(x) = ex 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
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