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Question
Examine the continuity of the following:
e2x + x2
Solution
Let f(x) = e2x + 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.
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