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प्रश्न
Examine the continuity of the following:
x2 cos x
उत्तर
Let f(x) = x2 cos x
f(x) is defined at all points of R.
Let x0 be an arbitrary point in R.
Then `lim_(x -> x_0) f(x) = lim_(x -> x_0) x^2 cos x`
= x20 cos 0
f(x0) = x20 cos 0
From equation (1) and (2), we have
`lim_(x -> x_0) x^2 cos x = f(x_0)`
∴ The 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 exist and is equal to the value of the function for all points in R.
∴ f(x) satisfies all conditions for continuity.
Hence f (x) is a continuous function in R.
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